course 09 curved bars

TRANSILVANIA UNIVERSITY OF BRASOV MECHANICAL ENGINEERING FACULTY

DEPARTMENT OF MECHANICAL ENGINEERING

ONLY FOR STUDENTS

STRENGTH OF MATERIALS - PART II Prof.dr.ing. Ioan Calin ROSCA

1

Course 9

Curved bars

9.1. Introduction

The beams with plane or spatial curved longitudinal axes are called curved bars. There are

considered two classes of problems:

a) initially curved beams where the depth of cross-section can be considered small in

relation to the initial radius of curvature.

b) those beams where the depth of cross-section and initial radius of curvature are

approximately of the same order, i.e. deep beams with high curvature.

c) The high curved bars bending theory was developed by Emile Winkler.

9.2. Initially curved slender beams

In this case the ration of 6...5h

R, where R is the curvature ob the bar and h is the height

of the cross section.

Let considered the curved bar from Figure 9.1. Under the complex load consisting of forces

Pk

Pn

Pq

N T

Mb G

R

O

Figure 7.1

and moments generic denoted with

...),...,,...,,2,1( qniPi , in the cross sections are

developed tensile and shear forces TandN,

and bending moments bM .

As is known the tensile force N generate

a normal stress uniformly distributed on the

cross section surface:

A

N , (9.1)

where A is the value of the cross section area.

The shear force T develop a shear stress

that is calculated with Juravski relationship:

z

z

Ib

ST . (9.2)



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The normal stress generated by the bending moment is found out with the classical Navier’s

relationship:

yI

M

z

b . (9.3)

9.3. Deep beams with high initial curvature

In case of a ration of 6...5h

R, where R is the curvature ob the bar and h is the height of

the cross section, the bars are considered to have high initial curvature (i.e. small radius of

curvature). In this case the bending stress has to be calculate with the theory developed by Emile

Winkler.

Figure 9.2

The theory is based on the

following assumptions:

The longitudinal axes are situated in a

single plane;

This plane is, in the same time, a

symmetry plane of the bar;

All the loads are applied in the same

plane that is the symmetry plane;

The cross section is considered to be

constant along the bar;

The material satisfy the Hooke’s law;

It is respected the Bernoulli’s

assumption which states that the

plane cross sections are normal to the

longitudinal axes before and after

deformation (the shape of the cross

section is changed, under loads, in a

neglected ratio and so one can

consider that remains the same);

It is neglected the compression

developed on radial direction by the

bending moment between the fibres.

It is considered a part of a plane

curved bar defined by angle d

(Figure 9.2).

There are made the following

notations:

1R - inner fibre radius; 2R - external fibre radius; R - distance from the curvature centre C to the

centre of gravity G ; r - is the distance to the neutral axis OO ; - is the curvature radius of a

fibre.



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Under the action of the bending moment bM the end sections (end edges) of the considered

element rotate one to the other one with an angle equal with d . To simplify the calculation one

can consider that only one end edge is rotated around the neutral axe.

There are made the following specifications:

The fiber that is situated on the neutral axe dose not changes its length. The

neutral axe divides the cross section in two parts: one where the normal stress

is positive (tensile), and the other one where the value of stress is negative one

(compression);

In the case of curved bars the neutral axe is not the same with the axe of the cross

sections centers of gravity (longitudinal axe) and results that it is necessary to

find out the position of neutral axe.

It is considered a fiber that has the length equal with ds that is situated at a distance y from

the neutral axe. The length of the fiber can be calculate, based on Figure 9.2, as:

dds . (9.4)

Under the bending load, the fiber length growth with a quantity ds that, according with

Bernoulli’s assumption, is:

dyds . (9.5)

As it can be seen in Figure (9.2), between the two quantities and y exists the

relationship:

ry . (9.6)

As was mentioned, it is considered that the material satisfies the Hooke’s law. This

assumption leads to the possibility to write, based on relationships (9.4) ÷ (9.6) the strain material

as:

d

dr

d

dy

ds

ds

, (9.7)

and the normal stress as:

yr

y

d

dE

d

dsrEE

. (9.8)

Observations:

From relationship (9.8) results that the variation of the normal stress , on the cross

sections, is represented by a hyperbolic function;

The highest values are developed in the fibres that are situated at the extreme edges of

the cross section;

In the neutral axe 0 yorr the normal stress is zero 0 .



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As it is known there are two relationships of equivalency written as:

,

;0

b

A

A

MdAy

dA

(9.9)

that, based on (9.8) become:

.

;0

2

b

A

A

MdAr

dx

dxE

dAr

dx

dxE

(9.10)

From the first equation of (9.10) one can obtain the geometric position of neutral axe:

0

AdA

rdAr

AA

, (9.11)

that leads to:

A

dA

Ar

, (9.12)

and the integral from denominator has different values according with the cross section shape.

From the second relationship of (9.10) is obtained the relationship of strain:

A

b

dAr

E

M

d

d

2

. (9.13)

The value of integral from relationship (9.13) is:

AeRArArAdAdAr

dArdA

r

AAAA

222

2

. (9.14)

Introducing (9.14) in (9.13) one can obtain:

eEA

M

d

d b

, (9.15)

that combined with (9.8) leads to the normal stress formula:

1

r

eA

M b ,



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or, yr

y

eA

M b

. (9.16)

Considering the geometrical notations there are obtained the following relationships for the

stresses developed in extremely edges:

inner edge: 1

11

R

y

eA

M b ; (9.17)

outer edge: 2

22

R

y

eA

M b . (9.18)

In the case of a tensile load that is superposed on the bending load, the total stress is given

by:

yr

y

eA

M

A

N b

. (9.19)

course 09 curved bars

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