-n model and prototype were performed using the matrix-dispacement method. large displacements of...

/-N AD-A016 778

FEASIBILITY STUDY ON DESIGN OF SELF-ERECTING AIRCRAFT SHELTER

Roy L. Johnson

New Mexico University

V_) C-

Prepared for:

Air Force Weapons Laboratories

October 1975

DISTRIBUTED BY:

National Technical Information Service U. S. DEPARTMENT OF COMMERCE

AD

A0

1Ö

77

8

315128 AFWL-TR-7 5-139 AFWL-TR-

75-139

A /1

1/

1/ 1/

FEASIBILITY STUDY ON DESIGN OF SELF-ERECTING

AIRCRAFT SHELTER

Roy L. Johnson

CERF, University of New Mexico

Albuquerque, New Mexico 87131

October 1975

Final Report

Approved for public release; distribution unlimited,

Reproduced by

NATIONAL TECHNICAL INFORMATION SERVICE

U S Department of Commerce

Springfield, VA 22151

AIR FORCE WEAPONS LABORATORY Air Force Systems Command

Kirtland Air Force Base, NM 87117

AFWL-TR-75-139

V

This final report was prepared by CERF, University of New Mexico, Albuquerque, New Mexico, under Contract F29601-74-C-0008, Job Order ILIR7425 with the Air Force Weapons Laboratory, Kirtland Air Force Base, Ní .» .'léxico. Lt Dennis Morrison (OL-AA, AFCEC/DEZ) was the Laboratory Project Officer-in-Charge. Captain Galen C. Bessert was the former project officer.

When US Covernment drawings, specifications, or other data are used for any purpose other than a definitely related Government procurement operation, the Government thereby incurs no responsibility nor an.v obligation whatsoever, and the fact that the Government may have formulated, furnished, or in any way sup¬ plied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise, as in any manner licensing the holder o¡ any other- person or corporation, or conveying any eights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto.

This report has been reviewed by the Information Office (01) and is releas¬ able to the National Technical Information Service (N7IS;. At NT15, it will be available to the general public, including foreign nations.

This technical report has been reviewed and is approved for publication.

DENNIS MORRISON Lt, USAF Project Officer

LOREN M. WOMACK Chief, Aerospace Facilities Division

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FEASIBILITY STUDY ON DESIGN OF SELF-ERECTING AIRCRAFT SHELTER ^

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Final Report

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Roy L. Johnson

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18 SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if necessary and identify by block number)

Shelters Aircraft Shelters Structures Air Mobility Structural Analysis Computer Code Erection Bare Base

20 ABSTRACT (Continue on reverse side If necessary’ and identify by block number)

This research examined the feasibility of designing and constructing an air- transportable, self-erecting aircraft shelter. Behavior of the shelter during erection and diserection was studied by field testing a 1/4-scale, segmented arch model of semicircular geometry, and by analyzing the behavior of a cir¬ cular segmental prototype structure. Analyses of the structural behavior of the model and prototype were performed using the matrix-dispacement method. Large displacements of the structure were accommodated analytically by (OVER)

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ABSTRACT (Cont'd)

use of a procedure of incremental displacement followed by an updatinq of structure geometry. A computer program (BOOTSTRAP) was developed to perform these analyses. Analyses showed that stable configurations of the 1/4-scale semicircular arch were achieved only when a single internal hinge was allowed in the structure. A satisfactory diserection procedure for the full-scale prototype was not achieved.

I0L

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SECURITY CLASSIFICATION OF THIS PAGEfWian Date Entered)

CONTENTS

Section

1

2

3

4

5

6

INTRODUCTION

MODEL TESTS

ANALYTICAL FORMULATION

MODEL STUDIES

PROTOTYPE STUDIES

CONCLUSIONS AND RECOMMENDATIONS

APPENDIX: PROGRAM BOOTSTRAP

PROGRAM STRUCTURE

INPUT INSTRUCTIONS

PROGRAM LISTING

Page

8

11

K

17

19 20

28

35

■

ILLUSTRATIONS

Figure Pm.

1

2

3

4

5

6

7

Model of Self-Erecting Aircraft Shelter 3

Joint-Locking Device 4

Hinge-Rotation Limiting Device 5

Partially Erected Model Configurations 6

Computed Configurations of Model 13

Prototype Configurations 15

Distribution of Bending Moments in Low-Profile Prototype 16

ii

aaiMiaiaMiHIIHM diimuk

SECTION 1

INTRODUCTION

BACKGROUND

With funding from the AFWL, faculty members of the USAT Academy successfully

demonstrated the feasibility of constructing a model of a self-erecting air¬

craft shelter. Thus, a model consisting of sixteen 2-ft-long prefabricated

modules connected to one another by a hinge mechanism was developed, ion-

trolled buckling of the structure at the hinges is obtained by applying a

moment to opposite ends of the horizontal beam column by a cable attached to

the ends. Since the structure resists bending in one direction and not the

other, it is gradually erected into a semicircular arch. This relatively

rapid and simple construction technique could make practical immediate full-

scale prototype application in support of highly mobile tactical air forces.

Therefore, feasibility studies on developing a prototype and design of such a

structure were undertaken at the Civil Engineering Research Facility (CERF).

OBJECTIVE

The overall objective of this research effort was to design an air-transpo^tabk:

aircraft shelter which can be erected in the field with a minimum amount of

equipment other than that provided with the shelter. Therefore, che feasi¬

bility of such a structural system with respect to the self-erection technique

was examined.

SCOPE

Field tests and analytical studies of a 1/4-scale model of a semicircular arch

and analytical studies of a full-scale prototype were conducted. An existing

computer program was modified and used to model the large displacement behav¬

ior of the arch.

1

SECTION 2

MODEL TESTS

The 1/4-scale model in its fully erected position is shown in figure 1. De¬

tails of the joint-locking and hinging mechanisms are shown in figures 2 and 3.

When received at CERF, all joints of the model were free hinged with the excep¬

tion of the first two or three joints near the ends; these were equipped with

gravity-actuated latches to lock the joints in their fully closed positions

(fig. 2b). Each joint was equipped with a bolt mechanism (fig. 3) which lim¬

ited the relative rotations of adjacent members when the joint was opened.

Positive or negative relative rotations of the members may be obtained by

adjusting this bolt. (A positive relative rotation is measured counterclock¬

wise between longitudinal axes of adjacent members.) However, the bolts pro¬

vided with the model (as received from the USAF Academy) permitted only neg¬

ative relative rotations of the members (fig. 3a). Therefore, the bolt mech¬

anisms were modified at CERF to permit positive relative rotations as well

(fig. 3b).

Several tests were performed on the 1/4-scale model to observe overall behav¬

ior during erection and diserection. From these tests and from f ilms-of tests

at the USAF Academy, it was evident that when no positive rotation of adjacent

members is permitted (model as received), the maximum loading on the structure

occurs when the first panel is raised (fig. 4a). To minimize the bending mo¬

ment in the arch, the joints were modified to permit positive relative rotations

of the members so as to minimize the span over which the weight of the struc¬

ture is carried (fig. 4b).

Observation of the erection and diserection of the arch with the modified

joints revealed the following facts:

(1) Cable tensions during erection were more constant in the early

stages although somewhat higher than those when no positive rotations

were permitted. As expected, a number of panels remained flat on the

pad, thus reducing the load which the semierected arch carried.

2

Fig

ure

1.

Mod

el

of

Self-E

recting A

ircra

ft

Sh

elte

r

(a) Opened Joint

n.25c

(b) Locked Joint

Figure 2. Joint-Locking Device

aHHHltttfattlilaiHi

(a) Before Modification

(b) After Modification

10c

Figure 3. Hinge-Rotation Limiting Device

J matiili

Su

bse

qu

ent

Con

figu

rati

on

1

c o 4-> H3 o

XJ o :r CD cn c

CD s- o CD CÛ

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c o 4->

fO o

“O o

CD CT> c

s- CD

«4-

Fig

ure

4.

Part

iall

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cted

Mod

el

Con

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ura

tio

ns

(2) The model assumed a symmetric mode during diserection; i.e., the

center joint opened and the structure symmetrically lowered. In the later

stages of diserection, the structure snapped-through into an asymmetric

mode and continued in this mode until diserection was completed. The snap-

through was violent and several joint pins failed. This snap-through was

also observed before joint modification; however, it was less violent be¬

cause of the restricted rotation of the joints.

To avoid snap-through, a modification was made to the cable arrangement to allow

the cable to be passed through a bracket at the second joint above the winch.

With this arrangement, the active cable tension was utilized to load the struc¬

ture asymmetrically, thus forcing it into an asymmetric mode for diserection.

Subsequent testing of the model with this arrangement showed that adequate con¬

trol of the configuration could be maintained during diserection and snap-

through of the arch could be avoided.

7

SECTION 3

ANALYTICAL FORMULATION

The structure was modeled analytically as a circular arch composed of straight-

line segments. Only one internal hinge was permitted at a time. Thrusts of

the arch were equilibrated by a cable tie capable of transmitting axial force

only Horizontal displacement of one of the supports was induced by lengthen¬

ing the cable tie and allowing the structure to displace under its own weight.

Displacements were applied in small increments to permit the use of a linear

stiffness matrix formulation. The geometry of the structure was updated at the

end of each displacement increment by adding algebraically the elastic displace¬

ments of the structure to the previous structure geometry. This permitted the

structure to undergo very large displacements without the use of a nonlinear

geometric formulation.

MATRIX FORMULATION

The s

were

tatic equations of equilibrium for a fixed configuration of the structure

{F} = [K]{u) (1)

where {F} and {ul are vectors of the generalized nodal forces and displacements,

respectively, and [K] is the assembled stiffness matrix of the structure formed

by appropriately summing the stiffness matrices for the individual elements of

the structure. An element is defined as that portion of the structure which

connects two nodes (joints). The formulation of [K] may be found in any stand¬

ard text on matrix structural analysis (ref. 1).

An equilibrium configuration is obtained from eq. (1).

{u} = [K]_1{F} (2)

1. Gere, J. M., and Weaver, W., Jr., Analysis of Framed Structures, Van Nostrand, New York, 1965.

iaiaMilllllHilfll

Here, the inversion of [K] is implied; but, in the actual computation a solu¬

tion was obtained by solving the set of simultaneous linear equations.

The vector {F} is a set of generalized nodal forces which consist of those de¬

rived from the gravity load, i.e., the weight of the structure, and those in¬

duced by initial support displacement. Accordingly, the vector {F} may be

written as the sum of the two parts; thus

í F} = IF] + {Fj g (3)

where the subscripts g and ui indicate gravity forces and forces due to initial

displacements, respectively. {F} is obtained by appropriately summing gener¬

alized nodal forces which are statically equivalent to the gravity loads for

each element of the structure.

Initial support displacements are transformed into equivalent generalized for¬ ces by

{nUi = [KjiuK (4)

where {u}^ is a set of initial displacements specified in small increments.

The stiffness matrix is reformulated at the end of each increment by algebra¬

ically adding the spatial coordinates of each node at the beginning of the in¬

crement to the corresponding nodal displacements occurring during that incre¬

ment. The updated geometry is then

Vi = xi f ui

Vl= ^i+ vi

2ifl = zi + wi (5)

where x, y, and z are spatial coordinates of a given node and u, v, and w are

nodal displacements in the corresponding directions. The subscript i indicates

the geometry at the beginning of the ith increment. The procedure is repeated

until the structure has been put through the full range of displacements desired,

9

The logic necessary to perform the computer analysis of the structure is pre¬

sented in the appendix.

10

SECTION

MODEL STUDIES

The behavior of the model during testing indicated that an analytical proce¬

dure capable of predicting equilibrium configurations and stresses in the arch

at various stages oc erection and diserection was necessary. Also, other than

semicircular configurations should be investigated in order to minimize struc¬

tural height while maintaining necessary work space. A computer solution

seemed most appropriate. Therefore, an existing University of New Mexico com¬

puter program for displacement analysis of elastic structures was extensively

modified for this purpose. (See appendix.)

The fully hinged 1/4-scale model was not successfully analyzed; however, modi¬

fications to permit analysis as well as to control configurations during dis¬

erection were investigated. With more than one joint opened in the arch, the

structure is a mechanism and only conditionally stable. The stiffness matrix

1's singular and a solution was not possible. Attempts to maintain small re¬

sidual bending and translational stiffnesses at the joints by using elements

with semirigid connections, such as those described by Gere and Weaver (ref.

I), also resulted in solutions which were unstable unless substantial residual

stiffness was maintained, but this resulted in structural models which were

too stiff when compared with the physical model.

In consultation with Captain Bessert and Lieutenant Morrison, AFWL/DEZ Task

Officers, it was decided to investigate the possible locking of certain joints

of the structure so as to maintain only one free hinge and permit unlocking of

subsequent joim; when the previously designated joint had fully opened and

regained bending stiffness. This locking and unlocking sequence could be de¬

termined in such a way that the structure could be forced to diserect in an

l. Gere, J. M., and Weaver, W., Jr. Nostrand, New York, 1965.

Analyi'da of Frame.*i Van

II

asymmetric mode. Subsequent analytical studies of the 1/4-scale model indi¬

cated that such a procedure was feasible.

Figure 5 shows the computed configurations of the model at two different sta¬

ges of diserection. Joint 3 was initially designated as a hinge and all other

joints were locked. The first configuration shows the geometry of the displaced

structure with joint 3 fully opened and with continuity restored at that joint.

Subsequent hinges were designated at joints 5, 7, 9, and 11 and formed sequen¬

tially as the right end of the arch was further displaced. The second config¬

uration shows joint 11 partially opened with all other joints locked. No joint

was allowed to open until the previously designated hinged joint had fully

opened. If this procedure is followed, the arch immediately goes into an asym¬

metric, stable mode for diserection.

12

i, UMipRUfifiJI iw- m ,«11,

13

Fig

ure

5.

Com

pute

d C

onfi

gura

tions

of

Mod

el

SECTION 5

PROTOTYPE STUDIES

^ -..il ACill /nP7 nffirprs to âccommoddt0 t-hG lã totype structure were establead by AF ' Z ^ ft-long panels binged

r;f,r2.; “rr:; r: «rr, r tow-profile configuration when the arch is fully erected.

hinge ^;;“”^;irntsh\r!nr,0ysis slid that the crown of ing suosequent incremental disp ^erection, resulting r,n: n: zirr :11::::1, the _ resuitr r a nearfy'syletric displacement -i.nation.^Howeyer, the analysis showed

that this configuration goes unstab

The low-profile configuration 11111:11:1 ri-:11: - -— 11:1:1:1:1 elastic deformation of the

points of contraflexure are approximately locate a

Ich moments are negative and tend to substan. q1ons. Subsequent displacements of the right end of the arch no

tially change this until large displacements of the crown occur,

the structure becomes unstable.

14

Figu

re 6

. Pr

otot

ype

Con

figu

ratio

ns

Fig

ure

7.

Dis

trib

uti

on

of

Ben

ding

Mom

ents

in

Low

-Pro

file

Pro

toty

pe

SECTION 6

CONCLUSIONS AND RECOMMENDATIONS

The analytical studies of the 1/4-scale model showed that, for a semicircular

geometry, the single-hinge concept was successful in producing a stable con¬

figuration during diserection. This was supported by observations of the be¬

havior of the physical model during the tests. From this, it is also con¬

cluded that the analysis procedure incorporated into the computer program used

in the analytical studies is adequate to predict the large displacement be¬

havior of the structure.

The inability to identify joint-hinging sequences which result in a satisfac¬

tory diserection procedure for the prototype structure suggests that the self-

erecting technique is not feasible with the low-profile configuration for the

following reasons:

(1) There is a possibility of instabilities of the arch during diserection.

(2) The size of the cable necessary for erection and diserection of the

prototype appears to be excessive (possibly 1-inch iiameter) with sub¬

sequent requirements of large size and high cost as well as weight of the

cable winching apparatus.

Analytical studies of other configurations should be made in order to fully

assess the feasibility of constructing a self-erecting arch.

17/18

W. üwiwwipuwTwm» ...

APPENDIX

PROGRAM BOOTSTRAP

A Special Purpose Program for the Large Displacement

Analysis of Circular, Segmental, Tied Arches

19

..- MtfiilÉltibÉÉiUÉÉINÉMi

PROGRAM STRUCTURE

BASIC PROGRAM LOGIC

A solution is obtained by a two-step procedure as follows:

(1) A displacement increment is applied to one of the end

supports of the arch and an elastic analysis is per¬

formed using the displacement method to determine an

equilibrium configuration.

(2) The configuration of the structure is updated and a

new elastic solution is obtained using the new con¬

figuration. This step stress relieves the structure

of the forces induced by the incremental displacement.

The force in the cable tie is thus a result of the

elastic deformation of the structure due to gravity

load only, using the configuration computed in step 1

as a reference.

Possible events occurring during step 1 are as follows:

(1) The designated hinge joint may close. Action: Restore

continuity and resolve before proceeding to step 2.

(2) The designated hinge joint may partially open. Action:

Proceed to step 2.

(3) The designated hinge joint may open beyond the maximum

value specified. Action: Iterate to find the incre¬

mental displacement which just causes the joint to open

to its maximum specified value before proceeding to

step 2.

Possible events occurring during step 2 are as follows:

(1) The designated hinge joint may close. Action: Restore

continuity and resolve.

(2) The designated hinge joint may partially open. Action:

Accept the solution and print results.

20

(3) The designated hinge joint may open beyond the maximum

value specified. Action: Restore continuity and resolve.

The program consists of a main calling routine (BOOTSTRAP) and the following

subroutines :

Name Principal Function

(1) PLOTXY

(2) INPUT

(3) STIFF

(4) BEAM

(5) LOAD

(6) DID

(7) MERGE

(8) BOUND

(9) SOLVER

(10) FORCE

(11) MODIFY

(12) GEOM

(13) BM

(14) ROTATE

(15) ANGCHK

(16) SUPCHK

Plots deformed structure.

Reads input data.

Calls element stiffness routine.

Forms element stiffness matrix.

Computes element load vector.

Computes member forces due to gravity loads.

Assembles structure stiffness matrix and load vector.

Applies boundary constraints.

Solves structure stiffness equations.

Computes member forces.

Calls modified element stiffness routine.

Updates structure geometry.

Forms modified element stiffness matrix.

Performs coordinate transformations.

Checks relative member joint rotations.

Checks joints for negative coordinates.

MAIN CALLING ROUTINE

BOOTSTRAP is the main routine which calls INPUT, STIFF, MODIt-Y, BOUND, SOLVER,

FORCE, GEOM, and PLOTXY. The principal parameters which control branching are

NONLIN, IUPDTE, IMOD, ITER, LASTIT, and ICONV. NONLIN is an input parameter

with a value of 0 or 1 depending on whether a linear or nonlinear problem, re-

spectively, is to be solved. (See section 2.) IUPDTE, IMOD, LASTIT, and ICO;«

21

are defined in subroutine ANGCHK or SUPCHK and may have values of 0 or 1. ITER

is an integer set to 1 or 2. When ITER = 1, an incremental displacement is im¬

posed on the structure. ITER is set to 1 at three stages in the solution:

(1) When an incremental displacement is imposed initially;

(2) During iteration to find the incremental displacement

which just causes a joint to open fully; and

(3) At the completion of a solution cycle.

When ITER = 2, the structure geometry is updated and a new solution is obtained.

SUBROUTINES

PLOTXY

PLOTXY calls a standard IBM/360 system subroutine called PLOT which plots up¬

dated joint coordinates on the line printer. Any other plot routine may be used

in lieu of PLOT for operation on another system. The y-displacements of the

structure may be amplified by using the factor AMPLE. (See section 2.)

INPUT

INPUT reads and prints control parameters and basic structure data such as mate¬

rial data, nodal point coordinates, member data, boundary constraint data, mem¬

ber release data, joint release sequences, and structure loading codes. The

half-bandwidth of the structure stiffness matrix is computed and printed in

this subroutine.

STIFF

STIFF forms the structure stiffness matrix. Only structures without member re¬

leases are handled in this subroutine. Member stiffness matrices are generated

in BEAM and transformed to global coordinates in a call to ROTATE before being

assembled in MERGE.

22

BEAM

BEAM forms the element stiffness matrices for only those elements without mem¬

ber releases.

LOAD

LOAD reads and prints concentrated joint forces, computes member Gravity loads

in a call to DLD, reads and prints member fixed-end forces from distributed

loads, and computes and prints member forces due to initial joint displacements.

Member forces are transformed to global coordinates in a call to ROTATE before

being assembled in MERGE to form the structure load vector.

DLD

DLD computes the equilibrating end forces for each member for the g-load speci¬

fied. (See section 2.) Forces are computed for the following conditions of

end fixity: fixed-fixed, hinged-hinged, hinged left, and hinged right.

MERGE

MERGE forms the structure stiffness matrix and the load vector by merging the

element stiffness matrices and the load vectors. The structure stiffness ma¬

trix is stored in array A(NUMEQ,MBAND), where NUMEQ is the number of simultane¬

ous linear equations (NUMEQ = 2 * NUMNP) and MBAND is the half-bandwidth of

the stiffness matrix. The structure load vector is stored in array B(NUMEQ).

BOUND

BOUND imposes boundary constraints on the structure stiffness matrix and on the

structure load vector.

SOLVER

SOLVER uses Cholesky's Method (ref. 2) for symmetrical matrices (Square Root

2. Ketter, Robert L., and Prawel, Sherwood P., Jr., n'în:'' c-

Engineering Computation, McGraw-Hill Book Co., New fork, 1969.

23

Method or Banachiewicz Method) to solve the system of simultaneous linear

equations. The stiffness matrix is reduced in a forward sweep only when mod¬

ifications to the structure occur, i.e., when the geometry is updated or addi¬

tional hinges are formed. The partially reduced stiffness matrix is used to

obtain solutions for each new load vector during iteration to find the incre¬

mental displacement which just causes the designated joint to open fully.

FORCE

FORCE computes member forces in global coordinates and transforms them to mem¬

ber coordinates in a call to ROTATE. Member forces are printed in the member

coordinate system. Element stiffness matrices are regenerated for use in FORCE

rather than written on and read from an external input/output unit.

MODIFY

MODIFY forms the structure stiffness matrix including members with releases.

Member stiffness matrices are generated in BM or in BEAM depending on whether

or not the member contains a release and transformed to global coordinates in

ROTATE before being assembled in MERGE.

GEOM

GEOM updates structure geometry by summing nodal spatial coordinates and cor¬

responding nodal displacements. Relative rotations of adjacent members at each

joint are computed using the updated geometry. A check on maximum and minimum

values of joint angles and minimum values of joint coordinates are made in

ANGCHK and SUPCHK which are called from this subroutine.

BM

BM generates stiffness matrices for members containing releases. Three types

of stiffness matrices are generated depending on the type of member release

specified: hinged-hinged, hinged left, and hinged right.

24

ROTATE

ROTATE transforms member stiffness matrices and load vectors into global coor¬

dinates. Member force vectors are transformed from global coordinates to mem¬

ber coordinates.

ANGCHK

ANGCHK checks relative member rotations to detect joint openings and closings.

The check is made on the angle (defined in section 2) for the interior joint

which has been designated as a hinge. Only three hinges, including supports,

are permitted in the structure. A hinge is modeled by specifying releases in

members joined at the designated hinge joint. A boundary constraint at the

joint which prohibits rotation of the joint about the z-axis is necessary to

reduce the singularity in the stiffness matrix resulting from zero bending

stiffness at the joint. The current and previous member releases are stored

in two arrays, IREL(NUMEM,2) and JREL(NUMEM,2); the first column contains the

previous releases and the second column contains the current releases. The

values in IREL and JREL are 1 or 0 depending on whether the member is or is

not released, respectively. These values change as joints open and close.

The principal parameters assigned in this subroutine and used to control branch¬

ing are ICLOSE, JCLÜSE, IUPDTE, IMOD, LASTIT, and ICONV. These parameters are

assigned values of 0 or 1 as indicated in the following table:

Set Flag ER

ICLOSE JCLOSE LASTIT IUPDTE IMOD ICONV

10 1 1 0 10

10 0 1 10 1

1 0 0 1 0 0 0

Event

Designated hinge joint closes on first iteration (JITER = 1) 4» < min

Designated hinge joint opens partially. 4

min <4 max

Designated hinge joint opens fully (JITER = 1 ). ¢) > ci>max. (Iterate to find new incremen tal displacement.)

25

table (Cont'd)

Event

Solution has converged. í =

tmax ■ Next joint in sequence is opened.

Designated hinge joint closes,

'min

Designated hinge joint opens partially. 4>min < 0 < ^max-

Designated hinge joint opens fullv. i > $ I U I ' J * T tmax

ITER ICL

1 0

Set Hag

¡CLOSE JCLOSE LASTIT IUPDTE ¡MOD ¡CpNV_

0 1 111

2 0

0 0 1

0 1 o

LASTIT = 1

1 sig- JITER is a counter incremented at the end of each solution cycle,

signals that one or more additional iterations are necessary. ICOT,

nais convergence of the iteration process when finding the incremental dis¬

placement which just causes the designated hinge joint to open fully, the

Regula Falsi Method (ref. 3) is used to find this displacement increment.

IUPDTE = 1 signals that the structure geometry is to be updated for the next

solution cycle. IMOD = 1 signals opening or closing of a joint not previously

opened or closed. When IUPDTE or IMOD is 1, the structure stiffness matrix is

reformulated for the next solution cycle.

SUPCHK

SUPCHK checks the updated y-coordinates of each interior joint for negative

values. This may occur during an initial displacement increment, whereupon an

iteration procedure is used to find the incremental displacement which just

causes the y-coordinate of the most critical joint to have a value of zero.

The most critical joint is taken to be the joint having the least value of

the y-coordinate algebraically less than zero. The Regula Falsi Method is

used in this iteration.

3 Carnahan, Brice, Luther, H. A., and Wilkes, J^mes 0., /.y < ■ - Methods , John Wiley & Sons, Inc., New York, 1969.

26

.11111,11111111(,111. 1111.1,..1 ».i. i¡iii : y- .l'iumii .1 11

Both ANGCHK and SUPCHK are executed at each solution cycle. The smaller of

the two incremental displacements computed in ANGCHK and SUPCHK is taken as

the next incremental displacement during iteration.

27

input instructions

STRUCTURE TYPE IDENTIFICATION (15)

Columns Variable Name

1-5 MTYPE

6-10 NONE IN

11-15 ITMAX

16-20 MAX IT

21-30 PHMAX

31-40 PHMIN

41-50 DX

51-60 AMPLE

Structure type identifier

1 = plane truss. 2 = plane frame. 3 = space truss. 4 = space frame.

Nonlinear solution control parameter

Any positive integer causes the program to branch to the nonlinear solution.

Maximum number of iteration cycles in a given displacement step

Maximum number of displacement steps

Limit of relative counterclockwise ro¬ tation in degrees of two adjacent mem-

Limit of relative clockwise rotation in degrees of two adjacent members

PHMIN

Negative Relative Rotation

Size of displacement increment in inches

Amplification factor to be applied to vertical displacements if required for plotting

23

TITLE (20A4)


I- 80 TITLE

CONTROL PARAMETERS (615)


I- 5 NUMNP

6-10 NBEAM

II- 15 N ST RUT

16-20 NUMAT

21-25 NRNP

26-30 NMREL

MATERIAL DATA (3E10.4)

Repeat NUMAT times (one card for each


1-10 E(IM)

II- 20 PR(IM)

21-30 RHO(IM)

Entry

Any alphanumeric identifier

Entry

Total number of nodes including supports

Total number of beam-type members

Total number of two-force members

Number of different materials

Number of constrained (supported) nodes

Number of member releases

different material ).

Entry

Young's modulus of elasticity

Poisson's ratio

Material density (wt/unit vol.)

29

NODAL POINT COORDINATES (15, 3E10.4)

Repeat NUMNP times (one card for each nodal point including supports).

Columns Variable Name Ln.trX

1 -5 NN

6-15 X(NN)

16-25 Y(NN)

26-35 Z(NN)

MEMBER DESCRIPTIONS (415, 5E10.4, 15)

Repeat NUMEM times (one card for each


1-5 ID

6-10 ND( ID ,1 )

11-15 NO ( ID ,2 )

16-20 MD(ID)

21-30 XA(ID)

31-40 ZI(ID)

41-50 YI(ID)

51-60 XJ( ID )

61-70 ALFA(ID)

Number of nodal point (must bo less than NUMNP)

x-coordinate

y-coordinate

z-coordinate

member). NUMEM = NBEAM + NSTRUT.

Entry

Member identification number (must be less than NBEAM + NS1RUT)

Nodal point number - node i

Nodal point number - node j

Material identification number

Cross sectional area (L‘)

Second moment of area about the local z-axis of the cross section (L" )

Second moment of area about the local y-axis of the cross section (L‘* )

Second moment of area about the local x-axis of the cross section (L4)

Angle of rotation in degrees of the principal y-axis of the cross section with respect to a vertical plane

30

71-80 MLC(IÛ)

Rotation

BOUNDARY CONDITIONS (13, II, 611, 6E10.4)

Repeat NRNP times (one card for each restrained node).

Columns

1-3

4

Variable Name

N

NTYPE(N)

5-10 IR(N,J)

11-20 UI(N,1)

Entry

Number of restrained node

Boundary condition type

0 or blank node is constrained against motion - no initial displacement spe¬ cified .

1 is initial displacement specified.

Type of constraint identifier

This is a six-digit integer, each digit of which is either 0 or 1 depending on whether mccion in the corresponding di¬ rection is constrained or unconstrained.

Examples

For a completely fixed node, IR(N,J) = 111111. This specifies that transla¬ tions and rotations are restrained in al 1 possible directions.

For a node restrained from translation in the global y-direction with all other motions free, IR(N,J) = 01000Ü.

For a node restrained from rotation in the globdl x-direction with all other motions free, IR(N,J) = 000100.

Specified initial translation in global x-direction (L)

31

- -TT"*-! ^

21-30

31-40

41-50

51-60

61-70

UI(N ,2 )

UI(N ,3 )

UI(N,4 )

UI ( N , 5 )

UI(N ,6)

Specified initial translation in global y-direction (L)

Specified initial translation in global z-direction (L)

Specified initial rotation about global x-axis (RAD)

Specified initial rotation about global y-axis (RAD)

Specified initial rotation about global z-axis (RAD)

If initial displacements are specified, a 1 must appear in the columns corresponding to the direction of the specified displace¬ ments .

MEMBER RELEASES (13, 211)

These data are required for members with initially specified releases only.

Repeat NMREL times (one card for each released member).

Columns

1-3

4

Variable Name

IM

MREL(IM,1 )

MREL(IM,2)

Entry

Member number

Release code for node i of member as specified in member data

Release code for node j of member

The release code is specified by a two- digit integer; either 1 or 0 is used depending on whether the i or j nodes of the member are released or not re¬ leased, respectively.

LOADING PARAMETERS (3I5,3F10.4)

Columns

1-5

Variable Name

NLND

Entry

Number of nodes with concentrated loads

32

mm

6-10 NLMEM

11-15 LGRAV

16-25 AX

26-35 AY

36-45 AZ

CONCENTRATED NODAL LOADS (I5.6F10.4)

One card per loaded node for each loading

omitted if NLND is zero.

Columns

1-5 NL

6-15 P(l)

16-25 P(2)

26-35 P(3 )

36-45 P(4)

46-55 P(5)

56-65 P(6)

MEMBER LOADS

Number of members having distributed loads along their length between nodes in addition to the member dead load

Member gravity load parameter

If 1, member dead load generalized forces are computed for a g-force as specified below.

Specified constant acceleration in the global x-direction (Not currently used)

Specified constant g-acceleration in the global y-direction

Specified constant acceleration in the global z-direction (Not currently used)

condition is required. Cards are

Entry

Node number

Component of concentrated force in globa x-direction

Force in global y-direction

Force in global z-direction

Component of concentrated couple about global x-axis

Couple about global y-axis

Couple about global z-axis

Variable Name

Two cards per loaded member (NLMEM members) are required for each loading con¬

dition involving members with intermediate loads Cards are omitted if NLMEM

is zero.

33

..

"W“1'f™ «ngpppoipqppqpr

FIRST CARD (2I5,6F10.4)

Coluinns Variable Name

1 -5 MM

6-10 I

11-20 P(l)

21-30 P(2)

31-40 P(3)

41-50 P(4)

51-60 P ( 5 )

61-70 P(6 )

SECOND CARD (5X, 15, 6F10.4)

Columns

1-5

6-10

11-20

21-30

31-40

41-50

51-60

61-70

Variable Name

J

P(7 )

P (8 )

P(9)

P(10)

P(H)

P(12)

Entry

Member number

Node number

Fixed-end axial force at node i

Fixed-end y-shear at node i

Fixed-end z-shear at node i

Fixed-end x-moment (torque) at node i

Fixed-end y-moment at node i

Fixed-end z-moment at node i

Entry

B1 ank

Node number

Fixed-end axial force at node j

Fixed-end y-shear at node j

Fixed-end z-shear at node j

Fixed-end x-moment (torque) at node j

Fixed-end y-moment at node j

Fixed-end z-moment at node j

3

Directions of Positive Fixed-End Forces

PROGRAM LISTING

BOOTSTRAP

PLOTXY

INPUT

STIFF

BEAM

LOAD

OLD

MERGE

BOUND

SOLVER

FORCE

MODIFY

GEOM

BM

ROTATE

ANGCHK

SUPCHK

Page

36

41

41

44

4b

46

49

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51

52

54

55

56

58

59

62

64

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-n model and prototype were performed using the matrix-dispacement method. large displacements of...

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