dr fp last hinge doc

7/31/2019 Dr Fp Last Hinge Doc

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Chapter 1

Plastic Hinges

Note Improved plastic hinging capabilities have been added to Dr. Frame

in this release, and the integration into the interface has been completed.

This documentation augments and partially supercedes the online infor-

mation regarding plastic hinging in the original Users Guide.

Dr. Frame provides you with the unique ability to model and observe plastic behaviorin structures interactively. This chapter presents Dr. Frames basic plastic hinging

functionality, some explanatory and cautionary notes, a set of short tutorial examples,

and some verification results. For a first reading, it may work best to skim the basic

operation section and then skip to the examples directly, returning to the more detailed

information later.

1.1 Basic Operation

The automatic installation/removal of plastic hinges is the key to modeling plastic be-havior in frames using Dr. Frame (Plastic hinges also can be created and manipulated

individually in a fashion similar to standard hingessee the online documentation for

information concerning working with plastic hinges manually.) The primary set of

plastic hinge commands is available via the Plastic Hinges popup in the Modeling

menu:

1


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2 CHAPTER 1. PLASTIC HINGES

To install plastic hinges at the ends of all members, use the Install Plastic

Hinges command (or type t). (As shown in the examples later, one generally

sets up appropriate hinge parameters prior to installing the hinges).

To remove plastic hinges, use the Remove Plastic Hinges command (or type

T (i.e., shift-t)). This in effect will repair the given structure.

To reset the plastic hinges, use the Reset Plastic Hinges command (type P

(shift-P)). This operation generally makes sense only if the loads have been

backed off below yield values.

Automatic plastic hinges will become activatedwhen the moment acting at thehinges location exceeds , in which and are taken from the prop-

erty settings for each member in effect at the time the hinges were installed. If

you change member properties, be sure to reset the hinges using a T/t unin-

stall/install sequence.

Plastic hinges display themselves as blue rectangles prior to yielding. Following

yielding, they display themselves as either solid red circles during loading or

open red circles during unloading.

Labels can be attached to plastic hinges to obtain a readout of the current plastic

rotation at the hinge. Use the Label Tool to click on the hinge in question, and

the label will be created automatically. Use the Units and Number Formattingmenu commands and dialogs to choose suitable label output. (Eventually such

labels will be able to store a sequence of values for purposes of plotting).

Note Labels attached to a plastic hinge will not become visible until

the hinge yields.


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1.2. NOTES AND CAVEATS 3

Figure 1.1: Plastic Hinge dialog box.

Double-clicking on an active/visible plastic hinge brings up the dialog shown in

Figure 1, which can be used to modify the hinges settings. Note that the post

yield stiffness is expressed as a multiple of the rotational stiffness of a hypothet-

ical beam segment with a length equal to the beams depth ( ). A value

of 0.002 for this parameter is a good starting value for steel members, combined

with a hinge offset of one member depth.

For purposes of setting default properties that all hinges will share, an alternateversion of the Plastic Hinge Dialog can be brought up using the Plastic Hinge

Parameters... command. This version of the dialog will let you set hinge loca-

tion offsets as a fraction of member depth, and you can specify a shared strain

hardening coefficient.

1.2 Notes and Caveats

Dr. Frame makes doing plastic analysis convenient, but there is no magic involved. The

following subsections explain how Dr. Frame does its plastic analysis, and what it can

and cannot do.

1.2.1 How it Works

Like virtually all nonlinear analysis programs, Dr. Frame exploits the fact that any

nonlinear analysis can be treated as a sequence of approximately linear solutions. In

fact, for the case of a structure with discrete, bilinear (or trilinear, or multi-linear) yield


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mechanisms and otherwise linear response, the behavior can be treated as exactly piece-

wise linear. In such cases, there is no need to control loading increment size beyondcapturing accurately the transition from one linear state to the next. In the present

context, these transitions correspond to the formation of plastic hinges.

To model hinge formation and the subsequent yielded response of a plastic hinge,

Dr. Frame uses plastic hinge objects (class in object-oriented programming parlance).

Each plastic hinge does the following:

Knows which member it belongs to, and where it is located on that member.

Knows what moment is necessary to cause itself to yield (or what combination

of moment and axial force will cause yield).

Installs itelf as a regular hinge combined with applied moments when active and

loading.

Keeps track of accumulated plastic rotation, , while yielding.

Keeps track of the previously applied moment at its location. This enables the

plastic hinge to determine moment increments from one load step to the next.

Watches each increment of rotation to detect unloading.

When unloading, installs itself as a locked up (rigidly constrained) hinge with

an imposed rotation jump. The corresponding constraint equation is of the form

.

These capabilities allow the plastic hinge objects to independently manage themselves,

and to take responsibility for most the details of the inelastic analysis.

The accurate determination of yield transitions requires communication between

the plastic hinge objects and the objects responsible for interpreting and applying user

actions. In particular, following each user action such as incrementing a load or set of

loads, or imposing a displacement, each plastic hinge checks to see if its yield capacityhas been exceeded. If so, it then reports to the action-controlling object the amount by

which its capacity was exceeded as a fraction of the action itself. The action controller

then backs off the action by the necessary percentage, which results in an applied incre-

ment just causing the given plastic hinge to reach its yield transition. The figure below

illustrates how this works:


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1.2. NOTES AND CAVEATS 5

Pi+1Pi

Mp

Mi

Mi+1

Pp

P

P'

..P

.

The plot shows the variation in moment at a particular hinge location as a function

of the applied load, . As the applied load increments from to , the moment

increments from to . For the case shown, this load increment causes the mo-

ment to exceed the plastic moment, . Since the relation between applied moment

and applied load is linear, it is straightforward to compute the percentage of the incre-

ment, , that should have been applied such that the moment would fall right at the

transition moment, . In particular, we have

With multiple hinges, it is necessary that each hinge check its moment following each

action/increment, and the minimum overall controls the increment used. Note that

this approach does not depend inherently on small increments.

There are various circumstances that can cause a nonlinear relation to exist between

moment and user action. Moving loads, changing geometry, etc., will lead to nonlinear

moment changes, in general. Including geometric nonlinearities will also result in a

deviation from the linear plot above. When the relation between moment and user

action is nonlinear, then it is necessary that one use relatively small increments.

1.2.2 Limitations

Plasticity brings with it a great deal of inherent complexity, and you can very easily

perform actions that will get Dr. Frame in an invalid or confused state. The main

issue is the fact that, unlike most geometric nonlinearities, plasticity introduces historydependence to a structure, and unlike a real structure, a computational model operates

in a context of discrete time. Any overly large discrete change in the structural model is

likely to lead to a bogus state. In general, then, work with small incrementing actions,

and if you need to make some large alterationsto a structure, uninstallthe plastic hinges,

back off the loads, make your changes, and then begin your analysis anew.


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Beyond issues associated with plastic analysis in general, there is the further issue

of the ever imperfect nature of software. As mentioned above, this version of Dr. Framehas several features/capabilities that are still in the works. Here are the known issues,

bugs, and missing features to date (send e-mail to [email protected] to

report other problems or suggestions):

Automatic hinge formation currently works only for end-of-span hinges. In-span

hinges need to be applied manually (i.e., by control-clicking with the plastic

hinge tool).

When 2nd-order analysis is enabled, the stability meter gives meaningless values

once the first plastic hinge has formed.

When including geometric effects (i.e., 2nd-order analysis is enabled), it is best

to hold off installing the plastic hinges until the structure has been loaded up near

yield. Before installing the hinges, use the i key to ensure the geometrically

nonlinear solution is fully converged.

The current plastic hinge yield criterion uses a very simple linear interaction

relation to account for axial load effect. The relation is of the form

in which is chosen acording to the level of axial load, as outlined in the AISC

LRFD specification.

Once a sufficient number of hinges have formed to cause a collapse, the structure

can not be restored by backing off the loads. This is actually realistic, but is dif-

ferent from the programs behavior following instability due to geometric effects

alone. To get back to a good state, heal the structure by uninstalling the plastic

hinges (T).

There have been a few changes in the program since the time the figures in this

document were madeyou may notice a few minor mismatches between whatyou see and what the figures depict.

The undo and redo commands have not been tested much yet with automatic

plastic hinges, so these commands might cause crashing problems. Always re-

member the golden rule of working with computers: Save early. Save often.


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1.3. EXAMPLES 7

1.3 Examples

The examples in this section illustrate the basics of doing plastic analysis. The best way

to go through the examples is to carry out the various steps yourself using Dr. Frame.

Example 1 The figure below shows a propped cantilever that we will use to consider

the behavior of simple plastic hinging.

0 k'

This configuration can be set up using the following sequence of operations:

1. Open a new file using the File:New command.

2. Use Modeling:Auto Beam... command to set up a simple beam configu-

ration. Use the default properties in the dialog.

3. Use the fixed support tool to change the left support by clicking on the

existing pin support. (If you are not sure which tool is which, refer to the

online Users Guide. Also, in addition to the standard pop-up tool tips on

the Windows platform, a brief help message appears in the status pane when

the cursor rolls over each tool button)

Now that geometry and support conditions have been set up, the next task is to

assign member properties and install the plastic hinges. For this example, the

only member property we will change is the plastic section modulus.

1. Use the select (arrow) tool to double-click on the beam.

2. Enter a new value in the Section Modulus edit box. Use to match the

results presented here.

3. Close the dialog.

4. Type t to install plastic hinges. Small blue rectangles will appear at the

ends of the beam.

For this example, we will apply the load via an imposed displacement at the right

support.


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1. Turn on displacement display by typing u (typing u again will toggle the

displacement display back off)

2. Choose the select tool, hold down the control-key, and click and slowly drag

the right support down. If necessary, drag a bit to the right as well to keep

the support at the end of the beam.

3. Release the mouse button and use the displacement display scaling button

to reduce the displacement scale. This will allow you to get higher moments

without the beam displacement diagram overlapping the moment diagram.

4. Control-click and drag the right support down again, and this time continue

dragging until the maximum moment reaches 45 k-ft and a plastic hinge

forms.

5. Continue to control-drag the right support downward, and note that the mo-ment remains constant at the left support.

6. Control-drag the right support slowly up, and note how the hinge display

changes to an open circle and the left support moment begins changing

again.

7. Continue dragging the support up until the hinge yields again the other way.

8. Return the right support to the original location by selecting it and typing

0 (zero), and note the residual moment and displacement.

9. Choose the label tool and click on the plastic hinge (be sure to click on the

hinge outline, or you will end up selecting the joint beneath the displayed

hinge). The label will indicate the plastic rotation across the hinge.

10. Return to the select tool and control-drag the right support through sev-

eral reversed yielding cycles, noting the accumulation of plastic rotation as

reflected in the label readout.

11. Type the T key to remove the plastic hinge and erase the effects of the

yielding history.

12. Back the support displacement off, if necessary, until the moment at the left

support drops below 45 k-ft. Type t to reinstall hinges, and you can start

a fresh elastic-plastic analysis.

Exercise a Attach a label to the right support so that you can read out the actualdisplacement you are imposing. Record moment and displacement values

as you do some cyclic plastic loading. Plot moment versus displacement.

Exercise b Repeat Exercise a, except before installing the plastic hinges, use

the Modeling:Plastic Hinges:Plastic Hinge Parameters... command to

bring up the plastic hinge info dialog. Use this dialog to set the post-yield


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1.3. EXAMPLES 9

stiffness to a non-zero value. Close the dialog, install the hinges, and do the

cyclic loading, recording values as before. Plot the final results.

Example 2 The figure below shows a simple frame with a vertical and a horizontal

load applied at the column-beam joint.

5.5 k

210.0 k

41 k'

To build this model yourself, do the following:

1. Starting with a new window, use the menu command Auto Frame... to

bring up the Auto Frame dialog. Set the number of bays to 1 and the bay

width to 15 ft, and then hit the OK button to close the dialog.

2. Use the arrow tool to drag a selection rectangle around the right column,

joints and support, and then use the delete key to remove these items.

3. Use the roller support tool to install a roller support on the beam similar to

the previous example: click somewhere in the beams span, and then drag

the support to the right end of the beam.

4. Since we will be applying loads with very different magnitudes, type theperiod key, ., to set fixed arrow lengths for the display of force values.

(This is a toggle: once you have applied your loads, you can type . again

to have the forces drawn to scale if you are curious.)

5. Use the point load tool to apply the vertical load. There are several ways to

accomplish this. You can click somewhere in the beams span and drag the


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load to the beams end, and then pull down to increase the loads magnitude,

or you can double-click in the beams span and set the load parametersdirectly using the ensuing dialog. If you click on the joint rather than one

of the members, you will get a joint load, which behaves a bit differently

when you drag it. For this problem it is easier to use member loads, so if

you made a joint load, select it, type delete to remove it, and then apply

another load starting with a click in the beams span.

6. Applythe horizontal load similarly by clicking in the columns span and set-

ting the loads magnitude and location appropriately using either the mouse

(click and drag) or the dialog (double-click and edit parameters).

7. Use the arrow tool to drag a selection box around the entire structure (not

including the moment diagram). Choose Object Info... (ctrl-i/cmd-i) from

the Info menu, and use the member properties dialog to set the beam and

column section modulus to as in the previous example.

8. Save this structure using the Save... command in the File menu.

Now that the structure has been set up, we will do some plastic modeling. We will

first do a purely piecewise linear analysis by ignoring bending-axial interaction.

1. Make sure the horizontal loads magnitude is such that the maximum mo-

ment displayed is less than 45 k-ft, and then install/enable plastic hinges for

each member by typing t.

2. Turn on displacement display by typing u.

3. Use the arrow tool to select the horizontal load (i.e., click on it), and then

use the cursor keys to increment the load. In particular, the up and down

arrows will change the magnitude of the load: in this case the down arrow

will increase the load (This load is pointing down relative to the members

orientation).

4. By default the load will increment by 5 times the current force display res-

olution, i.e., by 0.5 k when force values are displayed to one decimal place.

Holding down the control key while using the cursor keys will make the

increment match the display resolution, i.e. 0.1 kips in this case. Holding

down the shift key results in load increments 10 times the display resolu-

tion. For this particular problem, holding down the control key will give agood viewing increment.

Note The value display resolution can be modified for any set of

quantities via the Options:Number Formatting... menu com-

mand. This makes it possible to achieve very small increments

when desired.


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1.3. EXAMPLES 11

5. Continue increasing the load until the first plastic hinge forms. You may

note that there is an apparent hesitation in the load increase just beforethe hinge forms. This is due to the plastic hinge overriding the applied

increment as described in the previous section.

6. Continue loading until the second and final hinge forms. At this point the

structure becomes unstable, and the display will reflect this fact. Like a real

structure, backing off the load at this point does no goodyou are stuck in

an unstable state. To restore the structure, type T (shift-t) to invoke the

Remove Plastic Hinges command.

7. Use the cursor key to back off the load until the maximum moment again

drops below 45 k-ft. Re-initialize the plastic hinges by typing t.

8. Reload the structure up past the point of the first hinge forming, but shortof the second hinge forming. Use the cursor key to unload the structure,

taking the horizontal load magnitude all the way to zero. This shows the

residual moment and displacement arising from the plastic deformation.

9. Increment the load back up a bit, and then heal the structure by typing T.

We will now analyze the structure again, but this time we will account for the

moment-axial force interaction in terms of moment amplification.

1. Deselect the Resistance Factors On option in the Modeling menu.

2. Select the 2nd Order Analysis option in the Modeling menu. (You can

just use the comma (,) key to quickly toggle this option on and off). Thiswill cause a stability meter to appear as part of the display, and you will

note an amplification in the moment and displaced shape of the frame.

3. Toggle 2nd-order analysis off and on with the comma key to provide a quick

assessment of the relative importance of geometric effects.

4. With 2nd-order analysis turned on, look carefully at the value of the stability

meter, and type the i key. This key causes the nth order step command

in the Modeling menu to be invoked, which results in a further iteration

beyond the basic 2nd-order analysis. This particular frame is relatively sen-

sitive to geometric effects, so it takes several iterations to fully converge.

Note that the moment and displacement are not influenced much by the

additional iterations in this case, however.

5. Adjust the magnitude of the horizontal load such that the maximum moment

is less than 45 k-ft with the 2nd order effects included.

6. Use the i key to get a fully converged solution. This is an important step

when doing plastic analysis including geometric effectssubsequent solu-


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tions will all be based on the starting state at which the plastic hinges were

installed.

7. Type t to install the plastic hinges.

8. Select the horizontal load and increment as before.

Note It is more important that you use the control key in combi-

nation with the cursor keys to keep the increments small in this

case, since the problem is no longer truly piecewise linear. One

side effect of this nonlinearity is that you may think the program

has stopped incrementing when plastic hinges are about to form

keep hitting the cursor key: the hinge will eventually form. Lookat the figure in the previous section illustrating how load incre-

ments are determined in the vicinity of plastic hinge formation,

and consider what happens if the moment-load relation is nonlin-

ear with increasing or decreasing slope. This will help you un-

derstand what causes the apparent temporary freezing due to very

small increments being taken.

9. Load up to the formation of the second hinge, and the structure will become

unstable again. To restart the analysis do the following:

(a) Type shift-t to heal the plastic hinges and resurrect the frame.(b) Back off the load to bring the moments below yield.

(c) Use the i key to get a converged solution.

(d) Install new hinges by typing t.

Example 3 This example considers a somewhat larger frame with more complex load-

ing. Nevertheless, the only real difference between this and the previous exam-ples is the illustration of handling multiple loads. We start by constructing the

frame shown below:


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1.3. EXAMPLES 13

0.50 k/ft

0.50 k/ft

0.50 k/ft

5.0 k

3.5 k

2.0 k 39 k'

1. In a new window, choose the Auto Frame command from the Modeling

menu. Set the number of stories to 3 and the number of bays to 2, and hit

the OK button.

2. Use the arrow tool to drag a selection box around the entire structure, and

choose Object Info... from the Info menu. Set the section modulus to

, and clickOK.

3. This time we will give the hinges an offset from the joints. Therefore,

invoke the Plastic Hinge Parameters... command to get the appropriate

dialog, and enter the value of 1.0 in the location box. ClickOK to close the

dialog.

4. Type t to install member-end plastic hinges, and control-click with the

hinge tool to add in-span plastic hinges to the beams, if desired.

5. Use the distributed load tool and shift-click on each story. Holding down

the shift key causes the newly applied load to extend across all bays for a

given story.

6. Use the point load tool to apply horizontal point loads at each story level.

As before, it is best to apply the loads in the column spans, and then slide

them to the joints. (You may need to resize the window a bit and use the

panner tool to move things around so you can see the load magnitudes.)

7. Use the arrow tool and shift click on each load until all the loads are se-

lected.

8. Use the Group command in the Edit menu to group the selected loads.

Deselect the loads by clicking in a blank part of the window, and then click

on one of the point loads. Note that all the grouped loads are selected.


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Note Although all loads are selected, the load that was actually

clicked on will be used to define the load increments. All else beingequal, point loads provide more sensitivity than distributed loads,

so when point loads are available, use them to select the group.

9. Use the cursor keys as before to increment the loads (its not a bad idea

to use the control key to keep the increment size moderate). All the loads

are incremented together using proportional loading, with the increment

amount defined by the load that was used to select the group.

10. Continue increasing the loads and observe the various plastic hinges ap-

pearing in sequence. Remember that this is a load-controlled test, so once a

sufficient number of hinges form to create a mechanism, the structure will

be come unstable, and backing off the load wont fix it.

1.4 Verification

This section presents comparisons between published results for nonlinear frames and

analyses performed with Dr. Frame. The two comparisons presented here include one

experimental study and one analytical study. More extensive comparisons (which we

have already done) and detailed descriptions of analysis parameters and procedures will

be included in a future version of this documentation. The test files themselves will alsobe available from our web site at a future date. Note that the process of generating these

results uses the same procedures outlined in the previous Examples section.

1.4.1 Hybrid Frame

An experiment has been performed by Arnold et al., to investigate the inelastic be-

havior of a simple frame (Peter Arnold and Peter F. Adams and Le-Wu Lu, Strength

and Behavior of an Inelastic Hybrid Frame, ASCE Journal of the Structural Division,

94(ST1), 1968). This is a well known test, for which interesting behavioral phenomena

were found. The displacement based test yielded results that are useful for calibrating

various plastic analyses.Significant gravity and horizontal loads were applied to the frame, which caused

yielding influenced by geometric effects. During the test, the frame was successfully

restricted to in-plane movement for the first load cycle, which allows for 2D analysis.

The figure below shows the configuration as represented by Dr. Frame during an

analysis:


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1.4. VERIFICATION 15

5WF18.5, Ixx

10I25.4, Ixx

5WF18.5, Ixx

689 in-k

72.67 k

676 in-k

18.12 k

60.00 k

20.00 k20.00 k

60.00 k

116 %

5.53 in

Stability Ratio (EI = EI(P), phi's off): 3.76407e+95

The figure below presents a load-displacement comparison for Dr. Frame and the

experimental data:

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8

(in)

Experiment

Dr. Frame

Overall the match between the experiment and analysis is good (and comparable to

other published analyses of the same experiment).


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1.4.2 Vogel Frame

The ECCS (European Convention for Constructional Steelwork) has published the re-

sults of a six-story, two bay, plastic zone frame analysis. (See U. Vogel, Calibration

Frames, Stahlbau, 54(10), 1985, and U. Vogel, et al., Ultimate Limit State Calcu-

lation of Sway Frames with Rigid Joints, ECCS-CECM-EKS Publication, No. 33,

1984) The data have become a widely accepted benchmark used to calibrate structural

analysis programs with plastic hinging.

The figure below shows load-displacement curves from the above-referenced pub-

lications and Dr. Frame.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 4 8 12 16 20 24

Lateral Defl (cm) Top story

Vogel

Dr. Frame

Another comparison between these two analyses is shown in the figures below,

which illustrate the plastic hinging locations for the two analyses.


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1.4. VERIFICATION 17


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56.15 kN/m

56.15 kN/m

56.15 kN/m

56.15 kN/m

56.15 kN/m56.15 kN/m

36.25 kN/m36.25 kN/m

23.37 kN

23.37 kN

23.37 kN

23.37 kN

11.70 kN

23.37 kN

It should be noted that the Vogel analysis was based on a plastic zone approach,

which is a more involved and more accurate modeling technique than the discrete-hinge

approach currently built in to Dr. Frame. Nevertheless, these results indicate that one

can generate quite accurate results using Dr. Frame.

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