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ELA,STIO-PLAS~IO LOAD-DEFLEOTION, CURVE
O!ONSID~R,ING" 'S,~p',Q,ifD-oRDER, EFFEO~S ANJ) INSTABILITY
Lehigh University , ,Department of Oivil Engineering
,..:. Ii
i
(submitted to Dr.' A. Ostapenko in fulfillment', of the requlremen~s .toX-'·SPEOIAL;' PROBLEMS "I' IlJ,) OIVIL ENGINEERING - OE 406) . ,i-
De Gordon 'ollett, Jr~'
Fritz Engineering Laboratory Report
No. 273.57
" I
TABLE OF CONTENTS
ABSTRACT
1. INTRODUCTION
2. BACKGROUND DISCUSSION
2.1 Assumptions
2.2 First Order Elastic-Plastic Analysis
2 0 3 Secondary Effects
2.4 Instability'of EquilibJ?ium
2.5 Second Order Elastic-Plastic Analysis'
3. ITERATIVE PROCESS
1
2
4
4
5
8
9
10
14
3.1 General Discussion 14
3.2 Stable Range Iteration 16
3.3 Varification of Results and Discussion of Acc~racy 19- ---_ ... - - -
--.~-·---------------j.-4 Unstable Range Iteration 22
3.5 Influence of Axial Force on the Plastic Moment 26
4. SU1X1MARY AND CONCLUSIONS
5. APPENDIX
5.1 Nomenclature
5.2 Sample Problem
6. REFERENCES
28
29
29
31
62
ABSTRACT
A method' is here presented for a simplified approach to the
problem of determining the second order elastic-plastic_ load-deflection
curve, considering instability, for an unbraced, unsymmetrical multi- ~
story plane building fr~me subject to relatively large column loads.
No account is taken of the reduction in the plastic moment capacity
of the members which results from the presence of axial force. A brief
discussion is presented at the end describing how this effect might be
considered.
Much background material is included in an effort to give an
understanding of how this method was evolved.
The actual method is really two processes. T~e first covers the
solution of the problem when the frame is in the stable state. The
second process treats the frame after it becomes unstable, but only to
the extent that the point is located on the load-deflection curve where
the final hinge of the failure mechanism forms,
Finally, a sample problem using a simple portal frame is included
to show the reader an example of the application of the method. The
results of this problem show that for column loads equal to twenty (20)
times the beam load, the ultimate load capacity is equal to 6~%' of the
simple plastic mechanism'capacity.
- 1 ~
1. INTRODUCTION
This pape~ starts with the premise that the reader desires to have
before'.:.him the elastic-plastic load-deflection curve for an unbraced,
unsymmetrical~loadedplane frame and that he desires to consider second
order effects and instability. Such a frame could be subject to
relatively large column loads as might be found in the lower stories
of a multi-story 'building.
Conventional simple plastic theory formulates equilibrium on the
undeformed structure. This procedure is neither safe nor reasonable
where heavy vertical column loads exist such as in the lower stories
of a mUlt~-story building frame. Simply stated, the additional moment
generated because of the relative displacement of the column ends times
the vertical column loads-known as the ttP_L\tT effect--becomes too large
to be ignored.
As with all problems, there are many approaches to the solution
of this probleme The methods described in Refe (3) and (6) are
examples of how the problem may be attackede The procedure described·
in this report is an example of another approache It is based on the
assumptions that the relationship between moment and curvature is
elastic-perfectly plastid and that residual stresses, strain reversal
and the' effect' on the plastic moment of the axial and shear forces can
a-f"j~--be neglected (see Section 2.1 for full list of ;assumptions) e The
method presented is really two -methods,~ one deals. ~ithth~_ strlJ.q.tll~e.. in
the stable range, the other deals with the unstable range.
- 2 -
In the appendix an example problem using a fixed-ended portal
frame subject to vertical, horizontal and heavy vertical column loads
is used to il~u8trate the procedure and point out the importance of
considering the P-6 effects.
- 3 -
2. BACKGROUND DISCUSSION
2.1 ASSUMPTIONS
Certain basic assumptions are necessary before any further
discussion can begin. The introduction touched on some of these in
describing .what this report covered.
The following is the list of assumptions:
1. Material e lastic-perfe'ctly plastic.
cr yield'
Stress cr
M yield
Moment M
Strain €
CFig. 1)
Ca) Strain hardening neglected.
2. The relationship between moment and curvature elastic-
perfectly plastic.
Curvature ~
(Fig. 2)
Ca) Spread of plastification neglected.
3. Residual stresseS neglected.
4. Strain reversal assumed not, to take place. ,
5. Influence of shearing and axial forces on the plastic
moment neglected.
- 4 -
6. First order-equilibrium formulated on the 'lindeformed
member.
7. Seoond order-equilibrium formulated on the deformed
structure (P-~ effect).
8 . Instability inc,luded if it occurs prior to the
formation of mechanism~
9. It is considered a failure when a mechanism forms in
the structure.
10. Loading proportional.
11. Frame unbraced.
12. Either frame and/or loading unsymmetrical.
13. Loads act in a single plane and biaxial bending of
columns not considered.
14. Lateral bracing with simple connections prevents Qut
of-plane deformation.
15. All joints rigid with sufficient strength to transmit
the full plastic moment;,
2,2 FIRST ORDER ELASTIC-PLASTIC ANALYSIS
Neglecting the secondary moment and stability problems caused by
the P-/1 effect, i.e., t1ne additional moment cr,eated l"l'Jrten equilibrium is
, formulated on the deformed structure, the first order elastic-plastic
load-deflection curve is constructed by means of a step-by-step
procedure, The structure is as'sumed to have elastic regions which
- 5 -
control the deformations and localized plastic hinges. The load
deflection behavior is linear between the formation of successive
plastic hinges. The load-deflection curve can be constructed by
superimposing on the elastic load-deflection curve of the primary
structure portions of the elastic load-deflection curves of auxiliary
structures. The auxiliary structure is the resulting structure as
each plastic hinge is formed. The analysis requires a separate elastic
analysis after the formation of eac~ consecutive plastic hinge. This
leads to the generalized load-deflection curve shown below.
Load P
Ultimate load
Deflection 11
(Fig. 3)
The procedure for constructing the first order plastic load
deflection curve for a frame is as follows:
1. 'Analyze the structure elastically and draw the moment
diagram in terms of an unknown force 'PIe
2. Let the maximum moment equal the plastic moment Mpl and
solve for the value of Pl~
- 6 -
3. With the load and the moments known, the deflection can be
calculated and the point plotted on the load-deflection diagram.
4. Again analyze the frame elastically and plot the moment
diagram in terms of a new unknown force P2 but with a real
hinge inserted in the structure at the location of the
plastic moment, Mpl '
5. To determine the location of the second plastic moment, M 2. P
and the corresponding load'P2' add the moment diagrams from
step (2) and (4), then solve for P2. The smallest value of
P2 gives the correct location of the second plastic moment,
Mp2 (note that this value of P is the load increment, or
additional load, necessary to form the second plastic hinge).
6. Repeat steps (3), (4) and (5) until all of the plastic hinges
have formed, that is, until a mechanism has formed.
r't is important to remember when calculating a deflection corre
sponding to the' load necessary for the formation of a plastic hinge on
the load-deflection curve at some point (A~ that full continuity still
exists in the structure at this hinge location. This means that in, the
deflection calculations a real hinge can not be inserted in the
structure at point (A) until the next cycle.
- 7 -
2.3 SECONDARY EFFECTS
Structures which are subject to relatively high axial l~ads, such
as the lower stories a mUlti-story building frame, cannot always be
analyzed correctly without considering equilibrium of the deformed
structure.
The various theorems and methods for the analysis of indeterminate
structures which are dependent for their validity upon the applicability
of the principle of superposition constitute what is known as the
elastic theory. The principle of superposition requires two conditions
before it can be applied to a structure:
1. A linear relationship must exist betvleen stresses and
strains, that is, the material must follow HookeTs law.
2. The change in shape of the structure as loads are applied
may be neglected.
Violation of the first condition calls for a load-history analysis
such as nPlasticTheorytt$ Violation of the second condition requires
the use of ttLarge Deflection Theoryn, or, it may be referred to as
considering the nsecondary effects TT caused by th'e change in shape of
the loaded structure. It must be noted here that both the elastic
theory and the deflectiqn theory consider the structure to be elastic
but the latter condition requires that moments and forces be computed
for the final deflected position.
There are several degrees of "exactness Tt when formulating equilib
rium on the deformed structure. First, the least exact method, the
method used in this paper, is to consider the deformed structure. In a
- 8 -
multi-story frame building the secondary moment, or the 1t additional1t
moment which would be calculated in the columns would come from consi
"dering the relative displacement of the column ends times the vertical
component of the total axial load in the columns \ This is known as the
p-~ effect.
Secondly, it would be necessary to consider equilibrium as formu
lated on the deformed member as well as the structure. The third and
most exact method would be to add in the effect of axial shortening of
the members.
2.4 INSTABILITY OF EQUILIBRIUM
An unbraced unsymmetrical frame with relatively high vertical loads
on the columns, such as the lower stories of a multi-story building, may
be subject to instability before a' failure mechanism is formed. Under
these conditions' the second-order elastic-plastic load-deflection curve
will actually reach the maximum load point, (zero slope), before the
last plastic hinge forming the mechanism is developed. Any'increase in
deflection beyond this point must require a reduction 'in the load in
order to maintain equilibrium.
This phenomenon of instability can be explained by using the
following definition (Ref. 7, p. 407):
A system is said to be in a state of unstable equi-
librium if, for any possible small displacement from
the equilibrium configuration, upsetting forces will
arise which tend to accelerate the system to depart
even further from the equilibrium configuration.
The ttupsetting forces" referred to in the above definition are the
Ttsecondary effects n , specifically the p-~ effec,t, described in .Section
2.3 of this pa~er.
- 9 -
If one draws a generalized picture of the load-deflection curve
as would result from applying the iterative method described in this!
report to a frame structure we get the following showing the stable
and unstable range of behavior:
Pmax
Last hinge forms(Mechanism forms)
Load P
>Deflection 6-----L-------/:::?r!
Unstable Range~
Stable Range
(Fig. 4')
At this point it might be well to note that the value of the
maximum load is dependent on the loading sequence, however, if the
structure is not subj ect to, strain reversal in the plastified zones,
the value of the load at the formation. of the failure mechanism is
unique and independent of .the path of loading (Ref. 3, p. 13.10).
-
2.5 SECOND-ORDER ELASTIC-PLASTIC ANALYSIS
The second-order elastic-plastic load-deflection curve is
constructed'by means of a step-by-step procedure similar to the" first-
order curve discussed earlier. Again the load-deflection behavior is
considered as linear between the. formation of successive. plastic hingesu
The slope-deflection equation is used for solving the frame elastically.
The secondary, or p-~ moment, is introduced into the solution with a
- 10 -
condition equation by summing moments, including the p-~ moment, about
the base of one of the columns and then, by substitution, writing an
expression in terms of the lateral loads, the P-6 moment and the column
end moments. This is sometimes referred to as the Ttoverturning
moments Tt equation.
The prQcedure for constructing the second-order elastic-plastic
load-deflection curve using the slope-deflection method is as follows:
1. -Analyze the structure elast"ically, including the P-l1
effect, using slope-deflection equation in terms of an
unknown force Pl~
2. Let each resulting moment, M, equal the plastic moment,
~l' and solve the q~dratic equations for Pl , The root
giving the smallest absolute value of. P with a positive
deflection gives the location of the first plastic hinge
and the corresponding load Pl~
3. Draw the moment diagram.
4. Determine the deflection at the desired location and plot
the first point of the load-deflection curve g
5. With a real hinge inserted in the structure at the location
of the plastic moment, the second point on the curve will
be found ~
6. To simplify the calculations, only the incremental change
in moments and load are dealt with after the first hingeg
The first auxiliary' structure is now analyzed in a similar
manner as step (2). Note that the condition equation, i.e~,
the sum of the moments about a colUmn base, must deal only
- 11 - ·
with the moments caused by the additional deflection beyond
that at the first hinge.
7. To determine the location of the second plastic hinge and
the corresponding load P2' add the moment diagrams from
step (3) and step (6), then solve as in step (2).
8. Repeat steps (5), (6) and (7) until all of the plastic
hinges are developed, that is, until a mechanism has formed.
The solving of the simultaneous equations from the slope-deflection
solution gives answers in the form of a quadratic. Here we see'proof
of the non-linear relationship between the elastic analysis and the
s.econdary effects.
Of the two solutions·obtained from the quadratic equations for the
load P, the correct solution is the one which gives the smallest
absolute value for P and 'also gives a positive value for the deflection.
-A study was made of the second root but the exact physical meaning
~<:),~ ~~~ an,swer i.s. !!ot apparent 0 The __f?llow_~~g table gives the results
of a problem, Figo 5, solved by the slope-deflection method, that is .used--~ .... .' - -_ .._, .. ---""--_ .....~ .....-....-
later to compare with the proposed iterative method:
p
L
L ,P Lr ... ~l)"~~\1
~
(Fig. _5)
- 12 ,-
~
I
1
ROOT 1 ROOT'2P I ·~""--I- ~"---------.;... ----'6
-------+--------~---+_:-t-------+-------~o.---._.
+ -10.19 -
+ + 4.57 +
+ +19.12 I -~
I'II
+0.14944
+0.05555
+0 .03987-1(
+0.01588
+0.11166
+0 .01371
+0 .00248"it
(
+0.00875
+0.05022
+0.00588"i'C
+0.0009S oJ(
Hinge I
Hinge II
Hinge III
Hinge IV
+ - 6.83 I -+ - 2.95 i -
+ I + 2 .34 1.-I + ! +63 · 71 .! -
--......-----~--------........:ti-----t~'---·-· ---:r---.~ ..........-~.-"---~ 1 . ~
.I + I -15 . 82 1
! + I II - 3. 42 II + I + 3. 67 I
-"------i---------r--o-----f---+------+----.----
I I II + I - 4.15 !
-Ie Correct Answer
- '13 -
3 . ITERATIVE PROCESS
3.1 GENERAL DISCUSSION
Anyone using the slope-deflection procedure for constructing the
second order elastic~plastic load-deflection curve described earlier
will soon find it an extre~ely time consuming and arduous task even
for the very simplest of frames. In an effort to simplify the
procedure an approach is suggested based on assuming one or more of
the unknowns and then iterating until an answer within a specified
limit of accuracy is achievede
A brief description of the stable range iteration process is as
follows. This is the condition up to and including the point of
maximum load.
1. Assume a load P.
2. Find the moment diagram without considering the second
order effects.
3. Find the minimum load P required to produce the first
hinge and find the moment diagram.
4. Compute the deflection.
5. Find the mom~nt diagram for only the p-~ effect.
6. Add the momerit diagrams from parts (3) and (5)0
7. Repeat starting with step (3) until convergence.
- 14 -
Two additional points should be mentioned. First, by separating the
first order and second order moment determination, the non-linear part
of the calculations is bypassed. The second point is to consider the
general behavior of the iterative process. Af~er adding the first and
second order moment disgrams together, a linear relationship is assumed
between the load and the moment .. This tends to over compensate for any
error in the previous value of the load, The values of the load will
bounce bac:k and forth, sometimes greater, sometimes' less than the actual
value sought, but eventually converging to the correct value,
The iteration process for the unstable range is not. nearly so
straight forward as in the stable case. As shown later, the approach
used for the stable case will not work for the unstable case. The
method adopted is a simplified version of that described in Ref. (4).
This method gives the values of the deflection and load at the
formation of the last hinge only, i.e., the formation of a mechanism.
The points on the load-deflection curve at the formation of hinges
other than the last one cannot be determined by this method. A brief
description of the method is as follows:
1. Assume a failure mechanism 'and last hinge location.
2. Determine Pu, the ultimate load, by first-order rigid-
plastic, theory.
3, Compute the deflection.
4. Determine, Pu by virtual work on the deformed structure.
S. Determine the deflection.
--~ --.- ----··---6-.-~ Repeat starting with step (4) until Pu converges.
- 15 -
7. Checks:
Ca)
(b)
Plasticity condition (M < M )- p
Location of the last hinge.
If checks not satisfied, repeat computations for a new
mechanism and/or last hinge location.
3.2 STABLE RANGE ITERATION
As previously discussed, the stable range iteration is used to
determine the points on the load-defl~ction curve up to and including
the point of maximum load. If a failure mechanism develops at this
point, the curve is complete. If the failure mechanism has not
developed, the unstable range iteration that is described later must
be used.
The steps of the solution can very neatly be set up in table form
to minimize the work as is shown in the sample problem. It has been
bund convenient to treat the p~~ effect as a horizontal force equal to
P~6 where His the column height associated with the relative hori
zontal displacement of the column ends, ~,and P .is the vertical column
load.
The preliminary step required before beginning the iteration for
each hipge is to determine the moment diagram of the structure for
unit loads and unit P~6. The rest of the procedure is as follows:
First hinge:
1. ,Using the unit moments from the prelimlnary step, find the
first order load by proportion and reSUlting moment diag~am'
- 16 -
2 •
3 ."
4 .
necessary to develop the first plastic hinge. The location
of the first hinge will be at the point of maximum moment
for the unit load moment diagram. The load will also be
the minimum load necessary to develop the plastic moment.
Determine the horizontal deflection of the column ends, 6.
. p-~Determine the factor X, that lS, ~.
Using the moments from the preliminary step, find the
. resulting moment d~agram for X by proportion.
5. Add the moment diagrams from step (1) and (4).
6. Find the new load for cycle two by proportion between the
load from step (1), the maximum moment from step (5) andP2 M
the plastic moment, -- = PPI Mmax 1
7. Repeat steps (2) thru (6) until the change in the value of
the load between cycles is within the desired accuracy.
Second. hinge:
l~ Insert a real hinge at the point 6f the plastic moment .as
determined for the first hinge and proceed with the prelim-
inary step for the resulting auxi:liary structure.
2. Using the final moment diagram for the first hinge,
determine the amount of additional" moment, M -M. at each--~,-- p 1
critical location, i, that is, where~a plastic hinge might
form, required to form the plastic moment on the new structure .
- 17 -
3. Find the minimum incremental load necessary to form the
next hinge and draw the moment diagram similar to step (1)
for the first hinge.
4. Determine the incremental horizontal deflection of the
column ends, 6.
Since we are5.P-6Determine X, ~, for the second hinge.
working with incremental moments, loads, and deflections,
we must include the additional P-6 moment produced by the
load necessary· to form the first hinge being displaced
the additional incremental amount necessary to form the
second hinge.
6. Proceed as in step (4) of the fi~st hinge determination and,
find the moment diagram for X by proportion from the
preliminary step.
7. Add the moment diagrams from steps (3) and (6).
8.. Proceed as in step (6) of the first hinge determination
but with the proportion based on step (2) of the seconq
hinge determination. This must be done at each critical
location i. The critical location giving smallest value
of the incremental Ired change, P, is the correct location
of the next plastic hinge.
9 .' Repeat steps ('4)' trhru (8') as required for accuracy.
Additional hinges:
Proceed as in the second hinge determination but using the results
obtained from the previous hinge.
- 18 -
3.3 VERIFICATION OF RESULTS AND DISCUSSION OF ACCURACY
In order to assure that this iterative method does indeed give a
correct elastic-plastic load-deflection diagram and to help establish
accuracy guide lines a test, problem was solved by the slope-deflection
method. (See section on "Second-Order Elastic-Plastic Load-Deflection
Curve!T, Fig. 5 for sketch).
The follo~'ling diagrarns shO\£IJL.71g the moments \'lith the mecharlism
formed give a comparison of the results:
Slope-Deflection Solution Iterative Solution
61.00005xlO ft.lbs.
~
II !
6O.99981xlO ft.lbs.
= 0.05079 x l061bs .
= 1.95306 ft,~
Load P
Defl. fj.
J 6Jl.00008xlO ft.Ibs.j
i j! !
I ' j; 6I O.~9990xlO ft.Ibs.
II
6lxlO ft. Ibs .
/
6lxlO ft .Ibs .
Load P = 0.05079 x 10 61b8.
Defl. ~ = 1.95231 ft.
o .047_6M9__X_IO_,6_f_t_._1_b_S_.~~~~I_X~106ft.lbSI 0 .04674X_IO_6_f_t_._1_b_S_.~~~~~~~
(Fig. 6) (F~g. 7)
The results shown in the iterative solution were produced by
continuing the iteration until there was no ch'ange between cycles in
the last decimal place. The same problem was solved by limiting the
iteration to one cycle and the results were quite accurate (P = 0.05073
6· .x 10 Ibs., ~ = 1.95382 ft.).
- 19 -
The next step was to increase the loading on the column ends so
as to cause the frame to reach a condition of inst~bility before the
formation of the failure mechanism. The treatment of this condition
will be covered in t~2 next section in some detail but it is necessary
to mention it in connection with the discussion of the accuracy rules.
In' the description of the stable iteration process it was assumed
that the additional deflection produced by the B-~ effect was so small
that it could be neglected. This is 'not necessari~y correct. In
fact, there may not be any way of telling from the iteration process
that a structure has reached the unstable condition without including
this additional displacement. This was born out in the test pro~lems
where the structure' seemed to be stable unti~ the P-6 deflection was
included in the calculations.
A study was made 'of the two test problems described earlier and
,the following rules were developed to give an accuracy within
approximately 1.0% for the value of the load P:
1. For the first two cycles use the standard iteration
procedure as outlined in an earlier section.
2. Check the deflection at the end of cycle two, using the
final moment diagram, i.e,'the sum of the first order and
second order moments, and compare with ~ calculated from
the first order moments alone.
3. If t~e difference in step (2) is greater than 10.0%,
continue into cycle three using a modification of the
standard procedure.
- 20 -
4. The modification is to base the deflection calculation on
the final"moment diagram of the previous cycle as in step
(2) above rather than on the first order moments of the
current cycle.
5. Continue cycling until the change in the Load P between
cycles is less than 1.0%.
6. For all hinges after the first one these percentages can
be based on the first hinge 'values since the largest
proportion of the load and deflection will have taken
place at the format"ion of the first hinge. For -the first
hinge calculations the percentages are based on the
previous cycle values.
percent change in P
percent change in ~
= change between cycles x 100final value of P at first hinge
= change between cycles x 100final value of n at first hinge
The modification described in step (4) simply incorporates the
additional p-~ displacement into the iteration. The obvious question
is why not include the P-ll deflection in step (4) from the beginn~ng
instead of bypassing it for the first two cycles~ This was done in
several test examples and it was found that the number ,cycles
necessary for convergence increased two to three times with this.
additional complication~
- 21 -
The second cycle is used as the cut-off point in step (1) because
the results of the first cycle give a very high value of P which is
then corrected in the second cycle.
The 10.0% guide value in step (3) is highly ·empirical. The study
of the test problems actually showed that errors up to 5.0% in the
deflection still gave an accuracy well within the 1.0% limit for P.
Thus the 10.0% value was chosen for convenience. Certainly this area
could stand additional study and refinement.
3 .4 UNSTABLE RANGE ITERATION.'
vmen using the stable range iteration, if the load P keeps getting
smaller and smaller and seems to be heading for a limit of zero (0) and
the deflection ~ is getting larger and larger, the process fails to
converge. This means ,the structure has reached the unstable state.
The iteration process used for the stable range will not work when the
frame is in the unstable state and a different approach must be used.
Part of the difficulty is that now that the structure has reached
unstable equilibrium, the only way that equilibrium can be maintained
with increasing deflection is to decrease the load. If this was the
only problem, a simple modification of ,the stable iteration process
would be the soltuion. Unfortunately, if this approach is tried it
will soon become evident that convergence still is not going to take
place. The reason for this becomes clearer if one looks at a true plot
of the load-deflection diagram rather than the idealized plot of
assumed straight lines used for this discussion. Usually no hinge
will form at the point of maximum load and the idealized load-
- 22 -
deflection diagram will be a straight line between the last hinge to
form in the stable state and the first hinge to form in the unstable
state. The true plot, however, will rise up to a peak value of P and
then drop off again somewhere between these two h·inges.
I~ one is to write an equation for the incremental moment change
between the last hing~ in the stable state and the first hinge in the
unstable state, one would be faced with an interesting dilemma.
Knowing the additional moment required to develop the next hinge, an
equation can be written with P and ~ as the unknowns consisting of two
parts, the p-~ m~ment an¢ the first order moment due to a change in P.
If this equation is plotted for various values of P, one finds that at
P equal to zero, the deflection has a value. This would show up as a
straight line on the load-deflection plot and is inconsistent with the
physical ~ature of the problem. The on~y explanation for this dimemma
is that the equation of the ,load ....deflection plot is different on the
stable side of the maximum load point than it is on the unstable -side.
'since the iteration method used for the stable range makes use of this
type of equation, in order to use this method the formation of the
last hinge before the frame reaches the unstable state must be at the
point of maximum load or the moment diagram must be known when P is a
maximum. Therefore, this type of solution was abandoned in favor of
an approach utilizing a second-order rigid plastic solution~
The rigid plastic' iterative process, though, is one 'which
restricts the user to finding the load P and the deflection ~ at the
form'ation of the last hinge only. The basic procedure is a s'implified
- 23 -
version of the method described in reference (4). This method includes
the effect of axial force on the plastic moment and for this
discussion, as stated earlier, this effect is being ignored. The
p,rocedure is as follows:
1. Assume a failure mechanism and last hinge location.
2. Determine P , the ultimate load, by first-order rigidu
plastic theory. This is done by remembering the basic
ideas of applying. virtual wo'rk to the tTmechanism method TT
of solving a structure using rigid-plastic theory:
a. If a virtual displacement is applied to a system'
which is in equilibrium, the total work done is
equal to zero.
b. 'Yirtual'displ~cementis any;that is convenient to use with
:the assumption that the line of action of the forces does
"-:not change.
c. Virtual distortions are usually assumed as the
distortion of a linkage system the same as the
failure mechanism with no deformation between
points of rotation and with angular changes at the
locations of possible plastic hinges.
3. Find the horizontal deflection of the column ends, ~.
4. With the defl~ction in (3) determine the ultimate load,
p ,hy~virtual work on the ~eformed structure. The additionalu
consideration when using virtual work on the deformed
structure is that the virtual displacement is applied to the
structure with the mechanism already formed rather than the
,undeformed structure used in step (2).
- 24 -
5. Steps (3) and (4) are then repeated and a new~ Pu and ~
are' determined for each cycle until the change in Pu
between cycles is less than 1.0%.
6. There are two checks which must be made to determine if;
(1) the correct mechanism has been assumed; and (2) if the
correct location of the last hinge has been assumed:
a. The correct mechanism has been assumed if at no
point in the structure is the moment greater than
the plastic moment
M<MP
b. Once the failure mechanism is confirmed a check must
be made on the correctness of the assumed last hinge~
The general procedure is as follows (Ref. 8):
1. For a structure with R redundents, write R
simultaneous virtual work equations to
determine the values of 8., the concentrated1
slope changes at the plastic hinges.
(a) Introduce the unit moment, into the equilib-
rium structure in such a way that there is no
external work to contend with in the irtual'
work equations.
Wexternal = Winternal
o = ~ mi 8 i + ~ ~ SM m dx
m. = unit' internal moment on the1
equilibrium structure.
8. = plastic hinge rotation on the1
actual structure.
- 25 -
M = moment on the actual structure.
m = moment on the equilibrium
structure.
(b) The best way to eliminate the external work is to
put a double unit moment, (one for each slope
change) , on the equilibrium structure at the,
point of one of the plastic hinges and think
of it as an internal moment. Be sure that this
unit moment has the same sign as the plastic
moment.
2., . Solve these equations R + 1 times, (equal to the
number of hing~s required for a mechanism) . Set each
8. -in turn equal to zero, i. e. , equal to the lastl
hinge with no rotation.
3. Solution in which no 8. is equal to a minus valuel
give,S the correct last hinge (see Appendix 4-~2 for
sign convention).
3.5 INFLUENCE OF AXIAL FORCE ON THE PLASTIC MOMENT
In addition to causing instability, the presence of axial force
tends to reduce the magnitude of the plastic moment. The effect is
small in the case of small axial loads, and therefore in ordinary
portal frame columns any reduction is usually ignored. However, in
the case of multi-story structures, the resisting moment of the columns
can be reduced by axial load and the evaluation of the ultimate load
could include sueD consideration for a more accurate result. This
- 26 -
reduced moment is known as M and can be derived by standard procedurespc
(see Ref. 2).
In the previous discussion and in the example which follows the
·reduction of M to M has been neglected. A possible modificationp pc
in the stable iteration process to include this reduction would be to·
change from Mp to M at the end of the second cycle and repeat thepc
first two cycles. The discussion of the unstable iteration method
including M is covered in reference (4).pc
- 27 -
4. SUMMARY AND OONOLUSIONS
In summary, a method has been presented which gives a sim
plified approach to the problem of determining the second-order
elast1c~plastic l~~d-deflection curve including the problem of
instability. The method is a two-part iterative procedure - one
part for the s~able range 1 the other for the unstable range.
The stable range iteration separates the first order and sepond
order moment determination. Although mutually interdependent, the
procedure ,~s ,to determine the first order moment and resulting
defleciion an~ then'add the additional second order, or P-A
moment. This process is repeated until convergenoe within a specified
limit of accuracy.. is ,achieved G . By' ''using this type of procedur~, the--
time consuming non-linear part of the calculation~ is by-passe4.
In order to assure that the stable iterative procedure dO~$':
indeed give a correct, elastic-plastic load deflec·t~ion diagram, .0.::
tes,t problem was S,9lved by the s,lo.p:~-deflect1onmethod and b,y the
~terative proceduree A comparis,on of results shows the same'I'Ul:t:imate
load by' eit'her system ~nd only very small differences in the ·'··'final
moment diagrams. Also, as -'Ghe 'result of a .s'tudy of these test prob
lems, rough empirical rules of accuracy for the iteration were
developed •• - ; ~ I
After exploring several other approaches, the unstable range
it~ration procedure is developed from 2 method ~y Vog~l (see Refo 4)
and restricts the user to finding the values of the load and d~flection
at the last hinge onl~
- 28 -
In oonolu~1on, sinoe it has been shown that conventional s1mpl~
plastic theory, wh1c'q formulates ~qu111brlum on the undeformed
structure, is neither safe nor reasonable where heavy vertioal
column. loads ·exist and the results of the sample problem show that
for oolumn loads equal to only twenty (20) times the beam load, the
ultimate load capaoity is reduced to b4-% 'of the simple 'plastio
meohanism oapaoity, ~t is necessary ~o find some simplified method
of inoluding the soeond ,order:,,:,or P-A moment, wh.en dealing w1t~ the
plastio analysis' o£'.' heavily loaded oolumns.
- 28a -
( ,
5.1 Nomenclature
Text
E
H
I
M
Mmax
Mp
Mpc '
Myie'ld
P
Pu
R
W
X
11
€
8
cr'
cryield
~
s. APPENDIX
Modulus of elasticity
Column height of a story in a multi-story frame
Moment of inertia
Moment
Maximum moment
Plastic moment
Reduced plastic moment
Moment at- yield
Concentrated 'load,. axial load
Ultimate load
Number of redundents
Work
Lateral deflection of a story in a mUlti-~tory frame
Strain
End slope, rotation
Stress
Stress at yield
Curvature
- 29 -
Sample Problem
E
I
CDL
M
MADD .
MP.O.
Mmax"
Mp
MX-p
Pu
R
W
X
Z
6
cf'8
cry
Modulus of elasticity
Moment of inertia
Critical location i where a plastic hinge might form.
Length or height
Moment
Additional moment required to form plastic hinge
First order moment (neglecting second order effects)
Maximum moment
Plastic moment
Moment due to X
Concentrated load; axial load
Ultimate load
. Number of redundents
Work
P x 6/L
Plastic' modulus
Horizontal deflection of frame top
Deflection
End slope, rotation
Stress at yield
~30-
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6; . REFERENCES
1. American Society of Civil EngineersPLASTIC DESIGN IN STEEL, Commentary on, Manuals ofEngineering Practice Number 41, 1961
2. Beedle, L. S.PLASTIC DESIGN OF STEEL FRAMES, John Wiley and Sons, ,New York, 1958
3. Lehigh University ~
PLASTIC DESIGN OF MULTI-STORY FRAMES, FritzEngineering Laboratory Report ·No. 273.20, Bethlehem,Pennsylvania, 1965 ' .
4. Vogel, U.DIE TRAGLASTBERECHNUNG STAHLERNER RAHMENTRAGWERKE NACHDER PLASTIZITATSTHEORIE II. ORDNUNG, Heft 15,Stah1bauverlag, Koln, 1965
5 . Horne, M. R.STABI~ITY OF ELASTIC-PLASTIC STRUCTURES, Progress inSolid' Mechanics, Edited by Sneddon ~ Hill, Vol. 2,North Holland Publ. Co., Amsterdam, 1961
6. Parikh, B. P.ELASTIC-PLASTIC ANALYSIS AND DESIGN OF UNBRACED MULTISTORY STEEL FRAMES, Fritz Engineering Laboratory ReportNo. 273.44, May, 1966
7. Crandall, S. H. and Dahl, N. C~
AN INTRODUCTION TO THE MECHANICS· OF SOLIDS, McGraw-·HillBook Company, Inc., New York, 1959
8. Heyman, J.ON ESTIMATION OF DEFLECTIONS IN ELASTIC-PLASTIC FRAMEDSTRUCTURES, Frac. Inst. C~E., Vol~ 19, May, 1961, p. 39-60
Discussion by Martin, J. B.Proe. Inst. C.E~, Vol. 23, Oct,. 1962, p. 303~308
- 62 -
