Indirect Model Study

8/10/2019 Indirect Model Study

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educational section

INDIRECT METHODS OF

STRUCTURAL MODEL ANALYSIS

bY

J.

D.

DAVIES

University Co llege, Swansea

This article gives in two parts an elementary intro-

duction to the indirect methods widely used in

structural engineering and applicable also in other

fields. The first part introduces the basic theorems,

the second part in the next issue of Strain will dea l with

the laws of similitude, with model ma terials and with

experimental techniques.

1 Synopsis

Thepaper reviews ffietheory, methods and materials

used for indirect models for structures having linear

force-deformation characteristics. The main purpose

of these models is the dete rmin ation of influence lines

and influence surfaces for the generalised forces (axial

and shear forces, bending mom ents and torques) and

generalised displacements (axial and lateral move-

ments, rotations and twists) in prototype structures.

Unlike direct model methods, this method does

not have the actual loading on the structure re-

produced to scale, and this exp lains the term indirect.

Prototype quantities are ob tained from measurements

on the model in terms of the laws of structural simili-

tude. The assumption of linear behav iour considerably

simplifies the similarity requirements. A wide range

of

materials may be used for constructing the models,

the experimental techniques are com paratively simple

to p erform and expensive equipm ent is not required.

2. Introduction

Indirect models are used to determine influence lines

and influence surfaces for structures exhibiting linear

elastic behaviou r, i.e. structures in which the deforma -

tions are directly proportional to the loads. An

influence line is a graph giving the variation of some

linear response of the structure (force, moment, de-

flection or rotation) at a particular point as a unit

load (force or couple) moves along the structure. If

the unit load can mov e in two planes-as in shell

type structures-the influence line becomes an in-

fluence surface.

It is assumed that the principle of superposition

(see below) is valid for model an d full-size prototy pe.

The theory of indirect models is straightforward and

the testing procedures a re com paratively simple.

If influence characteristics are required for

genera-

lised forces

(axial forces, shear forces, moments and

torques) at particular sections, the procedure is to

cut the model at one of these sections, induce a

prescribed displacement, and interpret the results in

accordance with Muller-Breslaus principle. (See

Section 4.)

When influence characteristics are required for

generalised deformations

(deflections o r rotations) t he

model is deformed by a prescribed force and the results

are interpreted by recourse to the rec iprocal theorem.

(See Section 3.

Thus, for the first method the cutting and dis-

placing of the model requires suitable apparatus but

only deformations are measured. For the second

method the procedu re is simpler but the magnitude of

the force causing deformations has to be measured

also.

Indirect models are used for static or quasi-static

problems only.

3.

Linear Structural Behaviour

A large number of common civil engineering

structures remain stiff under load and the deforma-

tions are such that they do not m easureably affect th e

actions of the loads. Fo r o verall linear structural be-

haviour three conditions have to be fulfilled :

(i) The stress-strain relationships must be linear

(Hook es Law applies).

(ii) There are no cable o r stab ility effects, i.e., th e

deformations do not modify the stiffness of the

structure due to second order effects resulting

from axial tension o r com pression forces.

(iii) Th e deforma tions are small (no gross changes in

geometry under load)

These three requirem ents must be born e in mind in

the selection of suitable model m aterials, testing techni-

ques an d interpretation of results. Occasionally, it may

be

expedient to contravene requirement (iii) when it

is

more convenient to use large displacements if

precise measuring equipment is not to hand. The

page thirty-six Strain, October,

968


2/7

consequences of non-compliance with strict linear

elastic behav iour will be discussed in la ter sections.

Principle of superposition

The principle of superposition of forces and dis-

placements due to different loading systems by simple

algebraic summation is valid

if

the structure exhibits

linear behaviour. This principle, together with the re-

ciprocal theorem, forms the basis of all indirect

model analysis.

With regard to displacement, the required com-

ponent of the d isplacement at a po int is the projection

of

the actual deflection of the point on to the line of

action

of

the force acting at that poin t, i.e., the de-

flection affecting the work don e by the force (Fig. I.

P

Fig.

I .

The required component of the deflection is6,

namely the displacement affecting the work done

by P.

Influence coeflcients

The deflection

61

t some point 1 due to a load

P2

acting at some point 2 may be expressed in the form

61 = a12

p2

where

a12

is called the influence coefficient (or flexi-

bility coefficient) and represents th e change in deflec-

tion at

1

due to a change of unit load at 2.

Fig. 2 shows the load-deflection relationship for

a

point

1

on a linear structure subjected to a mon otonic

force system in which all the forces increase uniformly

at the same relative intensity.

Using influence coefficients, the deflection a t I may

be expressed as

I i 3

JI

Fig. 2. Monotonic forc e system.

Because the system is monotonic, the load ratios

Pz/Pl,

etc., are constant and all loads may be repre-

sented by

a

single load parameter, in this case

PI.

If the forces applied to the structure d o no t increase

in a constant ratio one to the

other

but in some arbit-

rary

way,

the load-deflection relationship at a point

will not be

a

straight line and some possible paths are

shown in Fig.

3.

Fig. 3. Possible load-deflection path s f o r a non-mono-

tonic for ce system.

However, irrespective of the loading pa th the total

net work done by the system of forces will be

1

8 - P 62

3)

When all the forces in the system have reached their

final values, the value of 61 will be the same irrespective

of the loading history, and the work done by P1 is

represented by the area of the hatched portion of

Fig. 3, always assuming linear behaviour throughou t.

The reciprocal theorem

1

a n d 2 .

Fig. 4a may represent a linear structure loaded at

Strain, October, 1968

page thirty-seveii


3/7

51

b

C

Fig.

4 .

The reciprocal theorem:

a12

=

a21

61 = al l PI

a12

PZ

6 =

a21 Pl a22 PZ

The total work done is

and is independent of the loading path.

Apply P1first and then apply Pz.The load-deflection

curves for 1 and 2 are shown in Figs. 4b and c. The

total w ork done for this sequence of loading is

,z- P 6 = all

P

a12 P1 p2

Equating (4) and 5 ) we find

which is one form of the reciprocal theorem.

1 1

1 5 )

2

2

a22 Pi

a12 = a21 (6)

-T

Fig. 5 . Bettis law: The work done by P1 in moving

through the displacement 61 caused by M2 equals

the work done by M2 in moving through the

displacement 02 caused by

PI.

Hence

Hence PI 61 = PlalzMz = M2e2 = M2a21P1.

~

Consider the simple structure shown in Fig. 5. We

first apply a load

P1

to point

1

and this will cause a

rotation e2=aZ1P1 t point 2. Remove

P1

and apply

a coup le M2 at point

2.

This will c ause a displacem ent

61=a 2 at point 1. Hence

a12=61/M2 and

a21

= /P1

Since

a12=a21,

it follows that

Pi

61 =M2 ( 6 4

Let us next consider two separate groups of

generalised loads: let PI represent the first group and

M

he second group. As before, we apply the system

of loads PI and then the system MII. Let QII be the

displacements caused by

PI

along the lines of action

of MI nd let

6 be

the d isplacements caused by MI,

along the lines of action of PI. The appropriate dis-

placements are those affecting the work done by each

component of the load groups-deflections for loads

and rotations for couples.

If

PI=P1 + PI/ +P1 . MI, =M2+M2/ +M2// .

s,

= 1/

s2// . = 2

2/

p ..

then com bining the work don e by each force compo-

nent we obta in in accordance with (6a)

(PI 61 +PI / 611 +PI/ 61// .

or

=(W +M2/e2/ +M2/82//

.

)

ZP161= ZM202 (6b)

This is known as Bettis Law and is simply a more

general form of the reciprocal theorem using the work

done by the two force systems. Thus if we expand

equation (6a) we get

P161=PI x a12 x M2=M2 x a21x P1=M2 , etc.

or in general,

1 61 = Z P2 62 for forces and deflections

Z M1 81 = Z

M2

e2for couplesand rotations (7)

Equations

(7)

represent the basis of influence line

theory. The displacement is considered positive if it

correspond s to the sense of the force.

1 61 =

M2

02 for mixed quantities 1

4.

Influence Line fo r Forces

Fig. 6a shows a model of a two-hinged arch.

Suppose it is required

to

find the influence line for the

horizontal thrust at the hinges. The restraint cor-

responding to the reactive force

H

is removed by

controlled cutting of the arch at point 1

;

t may be

imagined that the left-hand hinge is supported on

rollers permitting horizontal movements only (it is

page thirty-eighi

Strain , October, 968


4/7

important to note that indiscriminate cutting is not

permitted; only the restraint impeding motion of the

force under consideration along its line of action is

to be removed). The resulting structure is called a

primary structure (Fig. 6b) and is derived from the

original structure by removal of one restraint. All

subsequent operations are performed on the primary

structure.

/ p

b

/

c

d

Fig.

6 .

Mueller-Breslausprinciple: Diagram

c

shaded

area) gives the influence line for the horizontal

thrust H at the hinge

1.

The deflection

6

is considered positive if the

point

2

moves in a sense opposite to that of the

applied load.

a Original structure.

b Primary structure.

c Deformation of the primary structure due

to

displacement at

1

The ordinates of the

hatchedarea represent the influence line o r H .

d Deformation of the primary structure due to

a load P at

2

causing a deflection A at

1

H = 6 / A ) . P

Strain, October,

968

Two separate loading groups are now applied to

the primary structure and attention will be focussed

on the displacements at 1 and along an arbitrarily

chosen direction at an arbitrarily chosen point 2.

The primary structure is deformed by introducing

a small displacement at

1

along the line

of

action

of the reactive force H and the structure takes up the

shape indicated by the broken line. (Fig.

6c .

The

deflection at 2 is

6

= a21 H

= all

H

and that at 1 is

giving

Remove the displacing force at

1

and let the primary

structure deform under a load

P

Fig. 6d) applied at

2

causing a movement of equal magnitudein at 1. Then

equating the expressions for weifind

so

that with equation

8)

and with

a12

= a21 from (6)

A = a12 P

all H = a12 P

H 12

-

21

6

p - a l l - -n

giving

(9)

6

n

= - P

Since point 2 was arbitrarily chosen, it will be seen

that 6/A represent the shape and scale

of

the in-

fluence line for H due to a unit load P = l placed any-

where on the structure. Thus the hatched area in

Fig. 3c represents directly the influence line required.

This is the basis of Mueller-Breslaus principle

which may be stated as follows:-

For any linear elastic structure, if the restraint

corresponding to an internal or external force or

moment is removed and

a

corresponding trans-

lational or rotational displacement is introduced, the

deflected shape of the structure serves as an influence

line for the generalised force considered.

If the structure is statically determinate, it does not

have to be elastic for this principle to be valid.

Only the prescribed displacement

A

and the de-

formed shape of the structure are required; the

forces do not have to be measured. If the primary

structure is deformed by a prescribed displacement

acting in the assumed positive sense of the force under

page thirty-nine


5/7

consideration, the ordinates of the influence line will

be positive if they are movements opposite to the

sense of the applied loads.

If and P are equal to unity, 6 represents the

magn itude as well a s the shape of the influence line.

When Figs. 6 (c) and (d) are combined by the

principle of superposition, we return to the original

structure with a load

P

a t 2. If the applied load is a

couple, then

10)

where 0 is the rotation of the primary structure

at

the

point of application of M due to the displacement A.

Instead of measuring directly the component 6 of

the deflection along an arbitrary line of action it

is

sometimesmore: convenient t o measure the components

with respect to rectangular co-ord inate axes.

8

n

= - M

Nature of the cuts and displacem ents

The type of cut depend s on the particular quantity

to be determined : for axial forces the cut m ust permit

relative movements along the longitudinal a x i s of a

mem ber, for shear forces relative movements perpen-

dicular to the long itudinal axis, for bending mom ents

relative rotations about the point (a hinge), i.e., the

cut removes one restraint corresponding

to

the

quantity for which the influence line is required. Th e

induced displacement at the cut section must be such

that it only affects the work done by the quantity

under consideration. For example, when

a

hinge is

8

C

W

iiiuenck lines fo r internal reactive forces

a Original structure

b Influence line for the axial force at 1:

F =

6 / A ) P

Influence line for the shear force at 1:

S = 6 / A ) P

d Influence line f o r the bending moment at 1:

M

=

6 / a ) P

inserted and the primary structure deformed to find

the influence line for bending mom ent, there must be

no

relative axial o r shear displacem ents affecting the

net work don e by the axial and shear forces acting on

the section. There may be actua l movements provided

the work done by the eq ual a nd opposite forces is

algebraically zero.

The three possible types of displacements used in

planar frameworks are shown in Fig.

7.

These are

used to find the influence lines for the internal reactive

forces. For external reactive forces the displacem ents

depend on the nature of the support. Fig.

8

shows the

displacements appropriate to

a

clamped edge. If the

supports provide elastic restraint they can contribute

to the total work done when a displacemen t is intro-

duced, an d som e examples are indicated in Fig. 9.

5.

Influence Lines fo r Deformations

In this method it is not necessary to introduc e a cut

to form a primary structure: the deformations are

Strain, October,

968


6/7

M a

H

b c d

Fig.

8.

Types of displacement required to obtain the

influence lines fo r external reactive force s at a

clamped support.

a

Member with clamped support and with ex-

ternal reactive forces exerted by the support

on the member

b Displacement leading to influence line fo r H

c

Displacement leading

to

influence line fo r

d

Displacement leading to influence line fo r

M

.

Fig.

10.

Influence lines fo r the deformations

a

Original structure

d Deformation

of

the original structure due to

a unit load applied at an arbitrary position

2

c

Influence line fo r the vertical dejection at

1

d

Influence line fo r a clockwise rotation at

1

measured on the original (model) structure under a

known load acting at the point where the in ence

characteristics are required.

Consider the simple structure shown in Fig. 10a. It

is required to find the influence lines for the vertical

deflections and the planar rotation at the point 1 as a

unit load crosses the span. Apply a unit load P at an

arbitrary position 2: the deformed shape is shown in

r i d prop

C

d 'I

Fig.

9.

Dixere nt displacements fo r rigid and elastic

supports

2

Strain , October,

1968

page forty-one


7/7

Fig. lob. Remove

P

and apply a unit vertical load

at 1 in the assumed positive direction of the deflection

a t 1 (Fig. 1Oc).

From the reciprocal theorem

a12 =

a21

and the deformed shape (Fig. 1Oc) of the original

structure is the influence line for the vertical deflection

at 1 and represents the magnitude of the deflection

directly if = 1. If V 1 then the ordinate at the

arbitrary position 2 is

and it will be necessary to know

V

before

a12

can be

determined to give

similarly in Fig. 10d:

gives the influence line for the planar rotation at 1.

If the model is deformed in the assumed positive

direction of the d eformation required, the ordinate of

the influence line will be positive if it has the sam e sense

s the applied load. It will be observed that this is the

reverse of the sign convention used in the influence

lines for forces.

To be continued

6 = al2V

a12 = 6lV (1 1)

a12 = 6/M (12)

COURSES IN STRAIN MEASURE-

MENT AND ALLIED TOPICS

The following courses are being held at the South

Birmingham Technical College in association with the

Mid lands Branch of the B.S.S.M.

Short Courses

1969

18th Feb.-l8th March

Basic Strain Measurement

5

Tuesdays)

21st-25th April ,,

9

30th A p r . 4 t h May Three-Dimensional Photo-

Courses

of

10 Monday evening lectures

7th Oct.-9th Dec. Fat igue

13th Jan.-l7th March Measu rement Techniques

elasticity.

and their application to

Control

Full details and application forms for any of the

above courses can be ob tained on request from E. J.

Hearn, Senior Lecturer Mech.Eng. Dept., South

Birmingham Technical College, Bristol Ro ad S outh,

Birmingham 3

1

Strain, October, 968 page forty-two

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