﻿ Elastic and plastic flexural properties of generic shapes

Elastic and plastic flexural properties

Automatic computation of generic sections

Paolo Rugarli

[Costruzioni Metalliche, 4-1998]

Foreword

The search for better design and economic solutions leads more and more frequently to the use of non standard sections. The concept of standard section itself is loosing its meaning, due to the increasing number of available shapes.

Steel sections producers have usually sold their products together with small design manuals, detailed or not, listing all shape properties. In the meanwhile the manual number has grown, but none has solved the problem in full. The best effort made in Italy remains the Italian translation of the classic German text “Stahl im Hochbau”([1]), which gives a wide range of simple or composed shapes, and a large number of data, unfortunately not any more upgraded.

Text [1] is one of the vertex of the classic "manualistic" approach, thick volumes to be open in the most different cases, but it is common to believe that this approach should be updated in the light of today language, that is software. This need is not depending on fashion consideration: this need is to complete the effort made by those who wrote our classic texts, that is answering to a large number of problems. The designers and the producers today wish they can describe any section in an efficient and fast way, gaining all the section properties they need. A particular interest is directed toward those sections made up of a number of elementary sections (composed sections), and those sections cold formed, widely used by the industry.

This work explains the procedure followed by the Author to implement section computation in a very general way, within the SAMBA (Shape And Material Brisk Archive) project.

The objective of this work is to compute elastic and plastic flexural properties on a section or a set of sections completely general. The work will be focused on the numeric  and computational aspects featuring the problem, pointing out some of the problems to solve.

The section can be simple, composed, cold formed and can have holes: the procedure is completely general.

Description of a section using polygons

Generalities

In this work each section will be described as the set of a given number m of closed polygons, referred to a coordinate system (x, y). Each polygon can stand for a filled or for an empty region (a hole). Synthetically the section is so that

(1)

where Pi is the i-th polygon and hi is equal to +1 if this polygon is full, -1 if it is empty. It is straightforward that each curved side can be approximated by a given number of straight sides, if the number of straight sides is sufficiently high.

fig. 1

Each polygon Pi is described by n+1 points of the plane and by n sides, being the point Q1 coincident by definition with point Qn+1.

If this description is to have a meaning, it is necessary that no side of one polygon intersects another side (of the same polygon or of another polygon).

The points of each polygon are ordered from Q1 a Q n+1 in a counterclockwise way.

Integral computations

We are interested in the computetion of the following integral, defined in domain A internal to polygon P:

where p and q are two integers positive or null. Using Green's formula we have:

(2)

where polygon P is the boundary of A.

fig. 2

Therefore

(3)

If Qi has coordinates xi and yi and Δxi = xi+1-xi we can set, along the side Qi, Qi+1:

(4.a)

(4.b)

(5)

where λ is a nondimensional abscissa comprised between 0 and 1. Substituting (4)-(5) in (2) (3) we get

(6)

The defined integral in (6) can be evaluated numerically or in a closed formula. We introduce the short symbol (with three or four indexes)

(7)

where Qi is the starting point and p and q are the exponents of x and y, respectively. Thanks to (7) we can write

(8)

Some integrals are particularly useful. Precisely:

(9.a)

(9.b)

(9.c)

(9.d)

(9.e)

(9.f)

All these defined integrals are easily computable in a closed way. For instance:

and so on.

The result can be generalized if the section follows (1), that is if the section is made up of a set of m polygons, full or empty. In this case, the integral will be from point Qj of polygon i to point Qj+1 of the same polygon i, or shortly from Qij to Qij+1. Finally, generalizing (8) with a four index equation

(10)

that is each integral is reduced to algebraic sums.

Elastic flexural properties

Using the notation introduced we have:

(11.a)

(11.b)

(11.c)

(11.d)

(11.e)

(11.f)

From (11) it is possible to evaluate the section center G and the principal axes using ordinary methods. Let γ be the angle formed by principal axis u with axis x.

The distance of a generic point Qij (point j  of polygon i) from axis u, is

and from axis v, is

We now just set

(12.a)

(12.b)

finding the elastic section moduli.

Composed sections

One of the most frequent and interesting situation for the steel structures designer is the possibility to create a "composed" sectionassembling other elementary sections, so as to reach a given goal. In this field there are no rules limiting the possible choices, so it seens that any list, no matter how complete, is to be not complete enough. Often the need to join in a particular way the elementary sections depends on particular layouts, or on other reasons leading to an unrepeatable design need. The problem solution is to create a software able to let the designer free to join the shapes as he/she likes it. The computation procedure must be specialized to treat the problem efficiently.

Let us then call "composed section" Φ the reunion of an arbitrary high number f of elementary sections Θ. Each section Θ is referred to its Coordinate System (SC) (x, y) and has its proper principal axes (u,v) forming an angle γ with SC (x,y). We call instead (X,Y) the SC of the composed shape, and (U, V) its principal axes SC.

The position of each elementary section Θk in plane is described by three numbers: its center coordinates (Xk, Yk), and the angle αk of axis xk with respect to axis X.

First of all we note that the method previously described is still valid, since even a composed section satisfies (1), that is, it can be seen as a set of proper polygons. Of course, for the method to be applicable it is necessary that all polygons are referred to the same SC (X, Y), which can be done by imposing to all polygons i of section  k, Pki, a rototranslation depending on (Xk, Yk) and αk.

Software must update real time section computed data, while the user translates and rotates freely the elementary sections in the plane.

The overlapping tests assume a particular importance, since elementary sections cannot intersect. It is therefore important to assure that current choice of Xk, Yk and αk, that is the way the user has decided to move the current section in plane, does not violate these regularity condition. This is done checking that no polygon of currently displaced section, Pki , intersects the polygons of the other sections, and that no polygons of a section is contained or contains another polygon, picked by another section. Practically software must not accept as definitive situations where these conditions are not met, but just allow to "pass them over".

fig. 3

Besides the general method described, it is possible to use a direct method to compute second and first moments of the composed section, and to establish its principal axes position, starting from the same data of elementary sections. The formulae are the following:

(13.a)

(13.b)

(13.c)

(13.d)

(13.e)

(13.f)

Besides, setting βk=αk+γk

(14.a)

(14.b)

(14.c)

Equations (14) give the elementary section k second inertia moment, with respect to axes parallel to (X, Y) and passing through elementary section center. Substituting (14) into (13) we find the properties of the composed shape with respect to its own SC, as a function of the elementary section properties, of elementary sections positions (Xk, Yk) and of the rotation αk applied to them.

Once (13) are obtained, using general or direct method, it is afterward possible to compute the composed section center, its principal axes and its angle γ (angle between X and U). To gain the "center" second moment of inertia it will be sufficient to use the well known shift formulae. To compute section moduli the polygon description will anyhow be necessary, and the (12) evaluations.

Cold formed sections: some specializations

We define here "cold formed" a section that can be identified by an average line K and a constant thickness t. We assume that the line K is made up of straight and circular sides. Due to regularity conditions we set

K C1

That is the average line must be continuous with its first derivative. In this case the elastic properties can be computed using closed formulae. We set

(15)

where li is the generic side, straight or circular. We now write the contribution of each side to the relevant data. If li is straight, it is inclined of γ over reference axis x, has its center in Gi and is long bi, is is, easily

(16.a)

(16.b)

(16.c)

(16.d)

(16.e)

fig. 4

If li is a circular arc, we have, setting by definition zk= Rk-rk:

(17.a)

(17.b)

(17.c)

(17.d)

(17.e)

where xc and yc are the center coordinates, α and β are the two angle in figure, R and r are the external and internal radius, respectively.

The section properties are obtained summing up contributions of each side, for instance

where we use (16.c) or (17.c) depending on the side type, straight or circular.

To compute the section moduli W is anyhow necessary to transform the average line K of thickness t, into its equivalent closed polygon P, which is done bordering K of a thickness t/2, and transforming the circular sides in polygons with an appropriate number of sides.

Plastic flexural properties

Generality

The plastic section moduli computation has an increasing importance, due to the increasing popularity of limit state standards (EC3, BS, AISC, etc.).

Let us refer a section to its elastic principal axes (u, v). Given a generic plastic neutral axis (PNA) k (fig. 5) of equation

au+bv+c=0

where

this divides the section into two regions, a pulled region Ak+ and a compressed region Ak-. In the pulled region normal stress is +fy, in the compressed region it is–fy. Let us introduce the point function s(Q) thus defined:

s(Q) = sign(au+bv+c) = +1 if Q Ak+

s(Q) = sign(au+bv+c) = -1 if Q Ak-

At each generic plastic neutral axis k(PNAk) we get an axial force and two bending moments, that is

(18.a)

(18.b)

(18.c)

The tern fyΛk = {Nplk, Muplk, Mvplk}T is a point over the limit domain (one and not two because we assume that PNA is oriented). The vector Λk has as components the plastic moduli relative to the generic PNA chosen. Precisely we have:

(19.a)

(19.b)

(19.c)

Integrals computation

Let a polygon Pi , having internal domain Ai , be cut by an axis s. We will call Pi’ the polygon (equivalent to Pi) obtained adding to Pi the points found intersecting the sides of Pi with s.

fig. 5

If initially the points of Pi are (n+1), the points of Pi’ will be in general (n+1+r). The r new points stand all over s. We call Vij the points of the new polygon Pi’ (j goes from 1 to n+1+r), and we order the r new points found,  Ril, along s starting from the first toward the last, so that first and last are the most distant (fig. 6).

fig. 6

Given a couple of successive points Ril ed Ril+1, both laying over the polygon i, and a plastic neutral axis of equation au+bv+c=0, we introduce the function ηil(Ril) so defined (fig. 6):

ηil = sign(bΔuil - aΔvil)                if the middle point of segment RilRil+1 is inside Pi’

ηil = 0                                if the middle point of segment RilRil+1 is outside Pi’

This function ηil then equals +1 or –1, depending on the vector going from Ril a Ril+1 : if it has the same sign of PNA it equals +1, if has opposite sign equals -1, if the segment RilRil+1 does not belong to the domain it equals 0.

The need to introduce this function is purely informatic. It keeps into account two things: the first is that not every segment laying on s is effectively part of the section, and this must be understood by computer (for instance the segment Ri2Ri3 in figure 6). The second is that going from R1 to R2,R3 et cetera, you can run along PNA in its positive or negative verse, and this must be kept into account in evaluating contributions, which have positive sign only if they belong to the boundary of the pulled region, that is only if you run along the boundary in the verse of PNA (fig.5).

It can be shown that

(20)

where as usual

(21)

and similarily for Rilpq.

For a section made up of m polygons Pi, transformed into equivalent polygons Pi’ (getting new points Ril), we can then set, remembering (19) and applying (20):

(22.a)

(22.b)

(22.c)

The (22) tell how to compute limit moduli (and therefore limit actions) given a plastic neutral axis k. Note that values of hi are +1 if polygon Pi is full, -1 if polygon Pi is empty. Similarily sk(Vij) are +1 o –1 depending on the position of Vij with respect to PNA k (in pulled or compressed region), and that il is 1,-1 or 0. Therefore (22) are the sum with proper signs of a given number of integrals (9).

Search for plastic moduli

Among all the possible plastic neutral axes k, to which are associated the terns Λk, we are interested to the two axes PNAu and PNAv so that the two terns get, respectively

Λu = {0, Zu, 0}                (23.a)

Λv = {0, 0, Zv}                (23.b)

that is, to those PNA generating stress distributions in equilibrium with simple flexural actions.

Let us suppose we wish to get Ζu. This is done with an iterative process, trying to cancel Ζv and Npl.

Let us first consider how to cancel Npl.

Given a generic PNA slope angle ϕ , and written PNA equation in the form

vcos(ϕ)-usin(ϕ)-c = 0

it is easy to see that exists one and only one value of c, c=c(ϕ), and therefore one and only one PNA of angle ϕ, so that pulled region is equal to compressed region, that is so that

Apl = Ak+ - Ak- =0                (24)

This condition is necessary to have a purely bent section, that is to cancel Npl.

The c value corresponding to each generic ϕ can be found with an iterative method, using, for instance, the secant method (i is now the iteration index):

The error ε is computed as

(25)

Iteration is stopped when disequation (25) is satisfied.

At each c variation, leading to a translation of PNA at constant slope, we must compute the corresponding moduli using (22).

Let us now consider the Zv cancel.

To the value c obtained with a generic ϕ are mapped tern of the kind

{ 0, Zu(ϕ), Zv(ϕ) }

that is terns where Zv is not 0. The problem is to find the value of ϕ which cancels Zv, which is done with an iterative process. Let us set (i is the iteration index):

(26)

evaluating the error as

(27)

To each new ϕ a full iteration on c is done, to find the value c which meets (24). With the couple (ϕi, c(ϕi)) (22) are computed, the error is then evaluated through (27) and then a new ϕ is predicted with (26). Iteration is stopped when the (27) is satisfied.

Conclusion

fig. 7

The procedure here described has been implemented in the program SAMBA, and with it the elastic and plastic properties of complex composed and cold formed sections, like those shown in figures 7 (cold formed section with hole) and 8 (generic composed section) have been computed.

The generality of the method and its ready-to-implement features have allowed to solve the problem in a great number of cases, gaining the original goal.

fig.8

For instance, the study of composed shapes is done through dialogue of fig. 8, in which you can see as elementary sections (central rectangle, bottom) are added or removed (>>, <<) choosing them from a list (left rectangle).

The selected section (red in figure) can then be translated or rotated continuously (controls “X”, “Y”, “al”) or displaced "springly", searching for tangent-to-others conditions (buttons ->, <-, “su” [up], “giù”[down]).

Section data are upgraded continuously, while the plastic moduli can be computed at request, starting the double iteration described (“Computes plastic W”, in fig. 8).

Fig.9

Section data can then be printed to any device (fig.9).

Legend

α        angle between axis x and axis X, initial angle angle of circular side

β        final angle of circular side

γ        Angle between principal axis u and axis x

Δxi        by definition equal to xi+1-xi

ε        error in an iterative process

ϕ        Angle of slope of PNA over axis u

ηil        function of points Ril and Ril+1

λ        Nondimensional abscissa comprised between 0 and 1

Γ        boundary of A

Γ+        boundary of A+

Γ-        boundary of A-

Λ        plastic moduli vettore contenente i moduli plastici

Θ        Section made up by polygons

a        parameter of PNA equation

b        length of a straight side of a cold formed shape, parameter of PNA equation

c        parameter of PNA equation

d        distance of a point to an axis

f        number of elementary sections composing a new composed section

fy        yield stree

h        function establishing if a polygon is full or empty

i        index of a point over a polygon P, index of the polygons of Θ, iteration index, side index

j        index of a point over polygon Pi

k        index of the elementary section, index of the generic PNA

l        side of cold formed shape, index

m        number of polygons of a section

n        number of sides of a polygon

p        exponent iteger or null

q        exponent integer or null

r        internal radius of circular side, number of (new) points of P laying on s

s        straight line of PNA

s(Q)        function of point  Q

t        thickness of a cold formed shape

zk        Rk-rk

A        internal domain of a polygon, section area

A+        pulled region

A-Compressed region

C        center of the circle to which belongs a circular side

G        section center

H        common part between Γ+ and Γ-

I        second moment of area

M        bending moment

N        axial force

P        polygon

PNA        plastic neutral axis

Q        point of the plane if with one or two indexes, defined integral value if with three or four indexes

R        external radius of circular side, point of polygon P’ laying over s

S        first moment of area (static moment)

V        point of the plane belonging to polygon P’

Z        plastic section modulus

W        elastic section modulus

(x,y)        coordinate system of a section

(X,Y)        coordinate system of a composed section

(u,v)        principal reference system of a section

(U,V)        principal reference system of a composed section