Kuwait J. Sci. Eng.

29(2) 2002

Elastic and inelastic analysis of non-prismatic members using

®nite di€erence

SAMEER A. HAMOUSH*, MOHAMED J. TERRO**,

AND W. MARK MCGINLEY***
*
Dept. of civil and architectural Engineering, North Carolina A&T State University,
Greensboro, NC27411, USA.
**
Kuwait University

ABSTRACT
This paper studies the elastic and inelastic behavior of non-prismatic members subjected
to combined eccentric axial and transverse forces. The inelastic material behavior of the
member is taken into account using a bi-elastic constitutive relationship of the stress and
strains. This is to simulate the elastic and inelastic analysis as well as elastic perfect
plastic analysis. This is necessary to carry the model on various materials including
metals, concrete and composites. Based on the curvature-de¯ection relationship, a ®nite
di€erence technique is implemented to simulate the de¯ected shape of non-prismatic
members. An analytical computer model for the sectional equilibrium of the member is
developed. Exact solutions for six cases of strain in the pure elastic, elastic-plastic, and
pure plastic modes in both tension and compression are considered. The model relates
beam curvature to de¯ection by analyzing strain distribution at sections of the member
for a giver bending moment and axial force. The presented model provides a solution for
any non-prismatic beams, but the focus of this investigation is on tapered beams only.
Numberical examples are presented and validated against elastic solutions.

Keywords: Axial and transverse loading; beam-column; computer model; elastic

and inelastic. ®nite di€erence; non-prismatic.

1. INTRODUCTION
Beams that are deepened by haunches, to increase their moment resistance, and
non-prismatic columns, such as those supporting crane girders in industrial
buildings, are widely used in engineering practice. the elastic and inelastic
analysis of such non-prismatic members is well documented in the literature.
However, the inelastic analysis of non-prismatic members is limited to the use of
®nite element and equivalent systems methods. Furthermore, most of the work
on equivalent systems has been established for cases where the loads are applied
only in the transverse direction, and do not account for axial loads.
Ferdis (1956, 1966, 1973, and 1984), and Ferdis and Zobel (1958 and 1961)
166 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

developed the method of equivalent systems which used a double integration

technique to analyze prismatic members subjected to ¯exure only. The predicted
behavior of members under pure bending using this method compared
favorably with experimental results available in the literature. Ferdis and Keene
(1990) and Ferdis and Taneja (1991) further extended the equivalent systems
method to cover both prismatic and non-prismatic members in the inelastic
range. This method permits the replacement of the original member of variable
sti€ness with one of uniform equivalent sti€ness. By using this technique they
predicted the member behavior up to failure. Their study, however, did not
include members with axial loading.

El-Mezaini et al. (1991) used isoparametric plane stress ®nite elements to

conduct an investigation of the behavior of frames with non-prismatic members.
Their study showed that the conventional methods of analyzing these type of
structures lead to erroneous results. Therefore, they recommended the use of
frame analysis computer programs to model frames which contain non-
prismatic members.

Funk and Wang (1988) developed a numerical technique for the calculation
of a sti€ness matrix for non-prismatic members. This technique, coupled with
the ®nite element method, was used to obtain the de¯ections at nodes along the
member. The method, however, was not able to accurately predict the behavior
of the non-prismatic members in the inelastic range. Also, Resende and Doyle
(1981) and Mumuni (1983), working separately, developed ®nite element models
for the analysis of non-prismatic beams in the elastic and inelastic range. These
models are considered a reinforcement of the conventional ®nite element
method.

This paper dexcribes the development of a versatile and simple numerical

technique to study the behavior of non-prismatic beam-column members
stressed beyond their elastic limit. This technique uses a ®nite di€erence
approach over the length of the member coupled with force equilibrium and
strain compatibility equations applied across the section. Validation of
predicted member response against results obtained using other available
numerical techniques is presented.

The proposed model of this investigation was applied to reinforced concrete

beams (Shahid, 1998) and beams reinforced with FRP composite materials to
evaluate the de¯ection and the curvature along the cantilevered beams
(Hamoush et al. 2001). It was found that the proposed model predicted the
de¯ection with a reasonable accuracy.
 Elastic and inelastic analysis of non-prismatic members using ®nite di€erence 167

2. FORMULATION
In this section, the ®nite di€erence formulation and the inelastic beam-column
sectional analyses are described.

Finite di€erence formulation

Using a Taylor approximation for beam-column member, the ®nite di€erence
relationship between curvature and de¯ection is presented below (Allen and
Bulson 1980):
Wn‡1 ˆ h2 W00n ‡ 2Wn Wn 1 …1†
Where:
h = length of segments along the member, W = total displacement, n = station
number, and W00 = curvature.
Equation (1) provides the displacement at the end station when those at the
two previous stations are known, coupled with the curvature at the previous
station. It should be noted that higher order terms are neglected based on the
assumption of small de¯ections. However, equation (1) could be easily modi®ed
to include higher order terms by using a Runge-Kutta approximation.
1
The curvature (W00 ˆ , where  is the radius of curvature) is determined

assuming a linear strain distribution across the section at each station. This
linear strain distribution is de®ned by a top and bottom value of strain at each
end of the section. The top and bottom strains are calculated based on the
equations of equilibrium of the applied forces and the compatibity of strains at
each section. Ths boundary conditions for equation (1) are dependent on the
types of support used for the member under consideration.

(a) Simply supported ends

For beam-columns with simply supported ends, there are zero displacements at
both ends (see Figure 1).

Figure 1: Finite di€erence modeling of a simply supported beam.

168 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

Where:
W0 = initial displacement of unloaded member measured from the line joining
the ends of the center-line, e = eccentricity of the compressive load, and P =
applied load.
To apply the ®nite di€erence approach, a value for the displacement at station 2
must be assumed. Based on this value, the applied moment and axial force are
determined. Strain compatibility relations are then employed to calculate the
top and bottom strain values and the curvature of the section as shown in the
following sections. Having initial values for displacement at stations 1 and 2
(displacement at station 1 equals zero), equation (1) is then is used to determine
the displacement at station 3. The displacement at station 4 is then calculated
knowing the values at stations 2 and 3. This process is continued along the
member until the station at the end support is reached. If the end displacement
coincides with the actual support conditions, i.e. zero displacement, then the
assumed initial de¯ection at station 2 represents a true solution. Otherwise, a
new assumption is made based on the di€erence between the calculated and the
actual end conditions.
It should be noted that the applied moment at any station includes bending
due to the transverse loading in addition to the moment generated by the
applied eccentric load P. The method, therefore, accounts for geometric
nonlinearities by considering the P  e€ect. Furthermore, since the de¯ections
are measured from the center lines, initially curved members can also be
accounted for by this method.

(b) Fixed-end supports

In cases where the ends of the beam-column member are ®xed, zero values for
both the end displacements and rotations are imposed. To satisfy these
boundary conditions, an analysis is ®rst conducted with the end displacement
set to zero, as described for the case of a simply supported beadm-column. This
analysis will yield a set of displacement values at stations along the member. To
obtain zero end rotations, a ®xed-end moment is applied at each end of the
beam based on the value of displacement at the stations adjacent to the ends.
This ®xed-end moment is calculated as follows:
If Wn represents the displacement at the end, an imaginary station n ‡ 1 is
assumed with the following displacement value:

Wn‡1 ˆ h2 W00n ‡ 2Wn Wn 1 …2†

 Elastic and inelastic analysis of non-prismatic members using ®nite di€erence 169

But the rotation at station n equals zero …dw

dx ˆ 0† which can be numerically
formulated as:

Wn‡1 Wn 1
ˆ0 …3†
2h

Therefore,

Wn‡1 ˆ Wn 1

Substituting equation (3) into equation (1) leads to the curvature at the end:

2…Wn 1 Wn †
W00n ˆ …4†
h2

Using the values of the curvature at station n and the constant axial load P,
the top and bottom strain values are calculated. The ®xed-end moment is then
easily determined based on these strain values using the procedures described in
the later sections.
A second simple-end analysis is then carried-out with the calculated ®xed-end
moment added to the applied loads. A new set of de¯ection values are then
obtained along the member leading to an incremental ®xed-end moment which
is added in turn to the previous loads. The analysis is continued until the
incremental moment becomes within the tolerance.

(c) Simple and ®xed-end supports

In cases where the beam-column is ®xed at one end and simply supported at the
other, end displacements at the two supports and one end rotation are
restrained. The solution will be accomplished by performing an analysis with
simple supports ®rst, followed by a second one with an imaginary station
extended beyond the ®xed-end location to ensure zero rotation. The calculated
®xed-end moment is then added to the loads at the ®xed end and the analysis is
continued until convergence in a fashion similar to that of two ®xed ends
discussed in the previous section.

Inelastic Behavior of Beam-Columns

The inelastic analysis in the proposed model is based on the assumption that
plane sections remain plane during loading. Under applied loading, the unit
elongation of a ®ber at distance y from the neutral surface is described by
Timoshenko (1969 and 1976) and shown below:
170 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

y
"ˆ ‡ "0 …5†


Where, "0 is the strain shifting from

pure bending caused by the axial
force; the distance of the neutral
axis from the upper surface is
de no t ed by h1 ˆ "1 . a nd t he
distance between the neutral axis
and the lower surface is h2 ˆ  "2 ,
where  is the radius of curvature.
The magnitude of the applied
compressive force is given in the Figure 2. The shifted inelastic stress-strain diagram
following relation:
Z h1 Z "1 Z "1
bh
Pˆ b  dy ˆ b d " ˆ d" …6†
h2 "2
"2

where
ˆ "1 ‡ "2 , b and h are the width and the depth of the section
respectively.
The integral expression of equation (6) is equivalent to the area under the
stress-strain diagram. Equation (6) provides a relationship between "1 and "2 for
any given applied load.
The bending moment is given by the following relation:
Z h1 Z "1
Mˆb  ydy ˆ b 2 …" "0 † d ";
h2 "2

or
Z "1
12I
Mˆ …" "0 †d " …7†

"2

3
where I ˆ bh
12 is the moment of inertia for the section.

M can be derived for any set of values "1 and "2 . this is achieved by ®nding
the static moment of the area under the stress strain diagram with respect to the
centroid of the section.
M can be evaluated by performing the limited integration of Eq.(7) for any
set of maximum stain values "1 and "2 .
 Elastic and inelastic analysis of non-prismatic members using ®nite di€erence 171

3. MODEL DEVELOPMENT
For a given set of applied force P
and bending moment M, equations
(6) and (7) are the basic equations
employed for the determination of
the strain distribution across any
section. The top and bottom strains
are evaluated using the adopted bi-
linear elastic stress-strain model
shown in Figure 3. This idealized
stress-strain model describes the
material behavior in compression
and tension.
Six di€erent modes of strain can be
identi®ed in this stress-strain Figure 3. Bi-elastic approximation
model. these strain modes are of the stress-strain relationship
explained in the next section.

Modes of strain
The strain distribution across the section is assumed to be linear with an upper
strain "1 and a lower strain "2 . Therefore, depending on the values of "1 and "2
in the above strain distribution, six di€erent modes of strain can be identi®ed.
Three modes are in compression and the other three are in compression-tension
states. These strain modes are described below:
Mode 1 Compression - elastic state, j"1 j < "y and j"2 j < "y . the upper and lower
strains are in compression and below the yield strain. The stress and strain
distributions are shown in fugure 4.

Figure 4. compression - elastic state

172 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

Mode 2 Compression - elastic/plastic state, j"1 j > "y and j"2 j < "y . The upper
compressive strain is plastic and the lower compressive strain is elastic.
Mode 3 Compression - plastic state j"1 j > 3y and j"2 j > "y . both upper and lower
strains are in compression and exceed the yielding value.
Mode 4 Compression/tension - elastic state j"1 j < "y and j"2 j < "y . the upper
compressive strain is in tension and below the yield strain. The lower tensile
strain is also below the yield strain. The section behaves elastically.
Mode 5 Compression/tension - plastic/elastic, j"1 j > "y and j"2 j < "y . The upper
compression strain exceeds the yield strain while the lower tensile strain is still
below the yielding strain.
Mode 6 Compression/tension - plastic, j"1 j > "y and j"2 j > "y Both the upper
compressive strain and the lower tensile strain exceed the yield strain.
Using the mathematical formulation of P and M discussed in the previous
sections, a numberical model is developed to evaluate the upper and lower
strains ("1 and "2 ) for any section starting from the knowledge of the loads
applied. The technique implemented in evaluationg "1 and "2 may be
summarized as follows:
An initial set of values "1 and "2 is assumed. Based on these values, the axial
force and bending moment are calculated. An interative technique as shown in
Figures 5 and 6, is employed to converge to a set of values for "1 and "2 for
which the di€erence between the applied and the calculated forces is set below a
preset tolerance. This technique is described by Terro and Hamoush (1996).
This iterative technique is implemented in the computer model to calculate
the de¯ections and strains at each section from the knowledge of the axial force
and the bending moment.

Figure 5. Correction of "1 & "2 to minimize
P

 Elastic and inelastic analysis of non-prismatic members using ®nite di€erence 173

Figure 6. Correction of "1 & "2 to minimize
M

4. NUMERICAL APPLICATION
In this section, an example of a beam-column is analyzed using two cases of
boundary conditions: simple supports and ®xed supports. Di€erent loading
conditions are applied to both cases and the result obtained using the proposed
model is validated against those derived from classical solution methods.

CASE A: Simply-supported symmetric beam

Figure 7 shows the discretization of a typical symmetrically tapered simple beam
under combined axial and transverse loads.

Figure 7. Finite Di€erence discretization of a symmetric simply-supported beam

The beam under study is linearly tapered and symmetric with respect to the
mid-span. The cross-section at the ends is 279.46152.4mm (1166 inches) while
the middle section is 304.86152.4mm (1266 inches). The elastic and plastic
174 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

moduli, Ee and Ep , are preset at 179.3 GPa (26000 ksi) and 17.93 GPa (2600 ksi)
respectively. The beam has a length of 6.096m (20 feet) and is divided into 20
segments, 0.3048m (1 foot) long each.

The width b is constant and equal to 150mm.

Figure 8. Simple beam used to verify the proposed model

The proposed numerical model is used to ®nd the strain distribution and
de¯ection at the cross section of each segment. An elastic ®nite element solution
method (STAAD) is used for validation purposes. The beam is discretized into
20 two-noded beam ®nite elements with 21 nodes in total. Ranges of applied
axial force, 10kN to 10,160 kN (2.25 kips to 2284.2 kips), and for transverse
distributed loads, 5.3 kN/m to 73 kN/m (0.362k/ft to 5k/ft), are used in the
analysis.
The mid-span de¯ection of the beam is compared to the elastic solution
including P  e€ects. Table 1 shows a comparison between de¯ections
obtained using the proposed model and the ®nite element method with 3 cycles
and 9 cycles of P . The de¯ection results obtained using the proposed model
coincide with those of the ®nite element method while the strain values remain
in the elastic range. When the strains exceed the elastic range limit, the model
yields higher de¯ection values.
Table 2 shows the upper and lower face strains together with the de¯ection of
the midspan section using the proposed model. The behavior of the middle
section becomes nonlinear when the applied axial force reaches 10,000 kN
(2248.2kip) and the uniformly distributed transverse load reaches 55.8 kN/m
(3.82k/ft). When the applied transverse loads exceeds the value of 55.8 kN/m
(3.82k/ft), the method fails to converge to values of de¯ection and strains which,
in structural terms, can be trnaslated as the failure of the beam to support the
incremented applied loads.
 Elastic and inelastic analysis of non-prismatic members using ®nite di€erence 175

Table 1: The mid-span de¯ection of the beam analyzed

using the proposed model, Staad with 3 cycles and 9 cycles of P…†.
q(kN/m)
5.285 10.001 12.264 46.720 64.686 73.000 Model Used
P(kN)
1.392 2.692 2.997 11.811 16.002 18.009 Proposed Model
10 1.372 2.642 2.946 10.922 15.464 17.805 STAAD 3 cycles
1.930 2.794 3.150 12.319 16.761 19.050 STAAD 9 cycles
1.397 2.692 3.200 12.017 16.256 18.034 Proposed Model
100 1.372 2.642 3.048 10.973 15.494 17.780 STAAD 3 cycles
1.981 2.819 3.378 12.573 16.866 19.304 STAAD 9 cycles
1.422 2.769 3.429 12.573 17.018 19.050 Proposed Model
1,000 1.387 2.743 3.327 11.735 15.748 18.542 STAAD 3 cycles
2.007 2.921 3.607 13.208 17.043 18.796 STAAD 9 cycles
1.905 3.708 4.394 16.256 21.996 24.994 Proposed Model
5,000 1.854 3.607 4.343 15.748 20.726 24.536 STAAD 3 cycles
2.009 3.912 4.623 17.120 23.165 26.416 STAAD 9 cycles
3.099 5.994 7.188 31.496 Failed Failed Proposed Model
10,000 3.048 5.842 7.112 28.552 38.659 43.510 STAAD 3 cycles
3.266 6.325 7.569 30.894 41.834 47.092 STAAD 9 cycles

Table 2: The top and bottom strains, "1 & "2 , and the de¯ection obtained
at the mid-span using the proposed model
q(kN/m)
5.285 10.001 12.264 46.720 64.686 73.000
P(kN)
10 0.0610 0.120 0.130 0.520 0.700 0.790 "1  10 3
1.397 2.692 2.997 11.811 16.002 18.009 Mid-span de¯. (mm)
-0.057 -0.110 -0.130 -0.520 -0.700 -0.780 "2  10 3
100 0.078 0.130 0.160 0.530 0.730 0.810 "1  10 3
1.397 2.692 3.200 12.014 16.256 18.034 Mid-span de¯. (mm)
-0.041 0.092 -0.130 -0.500 -0.680 -0.770 "2  10 3
1,000 0.190 0.240 0.280 0.690 0.930 1.000 "1  10 3
1.422 2.769 3.429 12.573 17.018 19.050 Mid-span de¯. (mm)
0.060 0.004 -0.014 -0.410 -0.550 -0.630 "2  10 3
5,000 0.700 0.780 0.810 1.260 1.700 1.800 "1  10 3
1.905 3.708 4.394 16.256 21.996 24.994 Mid-span de¯. (mm)
0.530 0.460 0.430 0.129 -0.260 -0.350 "2  10 3
10,000 1.400 1.505 1.500 2.300 Structure Structure "1  10 3
3.099 5.994 7.188 31.496 Failed* Failed* Mid-span de¯. (mm)
1.100 0.970 0.920 0.094 "2  10 3
* The failure of structure is determined when the bifurcation at mid-span occurs.
176 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

A parametric study of the beam-column example is performed for a range of

axial forces and transverse loads and the de¯ection results are plotted in ®gure 9.
The axial loads were varied within the range of 10 kN to 10, 160 kN (2.25 k-
2248.2k) and the transverse load in the range 5.3 kN/m to 73 kN/m (0.362k/ft-
5k/ft). The plots in ®gure 9 represent the transverse loads versus mid-span
de¯ection for each of the applied axial loads. The results obtained using the
proposed model are compared to the ®nite element solution (STAAD) using
linear assumptions. The de¯ections obtained by the proposed model were found
to match with the ®nite element solutions when total behavior of all sections
along the beam is elastic.

Figure 9. The mid-span de¯ections of the beam-column example under combined distrbuted
and concentrated axial loads using the proposed model and STAAD (9 cycles).

When the strain in any section of the beam departs from the elastic range, the
de¯ection obtained by the proposed model exceeds that of the linear ®nite
element solution.

For the case when the transverse load is 65.4 kN/m (4.48 k/ft) and the axial
force is 10,160 kN (2248.2 kips), the obtained de¯ection in the proposed model
diverges at the point where P = 10,160 kN (2248.2 k) and q = 55.8 kN/m
(3.821 k/ft). The strain in compression at the middle section exceeds the yield
point and also the elastic de¯ection is under-estimated. When the axial force or
 Elastic and inelastic analysis of non-prismatic members using ®nite di€erence 177

the transverse load exceeds the above limits, the beam becomes critical and
failure by excessive de¯eciton may occur.

CASE B: Symmetric beam with ®xed supports

Figure 10 shows the discretization of a symmetrically tapered ®xed beam under
combined axial and transverse loads.

Figure 10. The Finite Di€erence discretization of a symmetric ®xed beam

The beam-column example described in case A is analyzed with ®xed ends

boundary conditions. The applied axial force and the transverse load have the
same limits as used for the simply supported case. Results summarized in Table
3 indicate that the beam remains in the elastic range. Table 3 shows the stations
de¯ections and the elastic de¯ection using the ®nite element method. All strain
and the de¯ection results coincided with those obtained using the ®nite element
method including the P  e€ect.

Table 3. Mid-span de¯ections (in mm) of the ®xed-beam

under combined distributed q and axial load P = 10,000 kN.

q (kN/m) Proposed Model STAAD 3 Cycles STAAD 9 Cycles

10.0 0.889 0.838 1.041
46.72 4.013 4.005 4.013
64.87 5.359 5.425 5.428
73.0 6.096 6.096 6.109

The top and bottom strains, "1 and "2 , at the ends of the beam are determined
using the following relations:
178 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

- Form W00n a relationship between "1 and "2 can be established as:

h ˆ h1 ‡ h2  …"1 "2 † …8†

- Equilibrium of forces gives:

Z "1
Pˆ b d" …9†
"2

Using equations (8) and (9), the strains "1 and "2 can be calculated.
- The equilibrium of moments, using the calculated values of "1 and "2 , gives a
®xed moment as follows:
Z "1
12I
Mˆ …" "0 † d " …10†

"2

5. CONCLUSIONS
The developed technique is based on establishing a moment-curvature
relationship for sections subjected to a combined axial force and bending
moment. Material and geometric non-linearity are considered in the method.
Elastic-plastic constitutive relations are adopted to acount for the material non-
linearity. The geometric non-linearity is solved using a ®nite di€erence approach
to predict the updated de¯ected shape of the beam-column in a step-by-step
interation procedure. This technique can be used to study non-prismatic beam-
columns. Even though the investigation is carried out for beams with an axis of
symmetry about the middle point, the model is capable of solving non-
symmetric cases. A series of de¯ection values of nodes along the beam-column
are obtained. These de¯ection values describe the ®nal de¯ected shape of the
element under study due to nonlinear behavior. The results from a number of
cases were validated against those obtained from an established ®nite element
structural program (STAAD). The method which was the subject of this
research is accurate and fast compared to the ®nite element approach. This
technique can be used to ®nd the buckling behavior of the beam-columns which
were under investigation by the authors.

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Submitted : 6 December 2000

Revised : 2 January 2002
Accepted : 11 February 2002
180 Sameer A. Hamoush, Mohamed J. Terro, and W. Mark McGinley

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