E”i&3acements

44 Two-dimensional problems in elasticity

Fig. 2.5 Bending of an end-loaded cantilever.

the cantilever beam shown in Fig. 2.5 is

Bxy3 f$ = A x y + - 6 where A and B are unknown constants. Hence

b (ii)

I

T x y = - - = - A - -

$4

BY2

J

axay 2

Substitution for f$ in the biharmonic equation shows that the form of the stress function satisfies compatibility for all values of the constants A and B. The actual values of A and B are chosen to satisfy the boundary condition, viz. ^{T,~ =} 0 along the upper and lower edges of the beam, and the resultant shear load over the free end is equal to P.

From the first of these

BY2 b

at y = f-

2 2

T~,, = -A --= 0 giving

A = - - - Bb2 8 From the second

bI2

-

J

^{T , ~ dy = P} (see sign convention for ^7-yy)

-b/2

or

- dy = P

-b/2

2.6 Bending of an end-loaded cantilever 45

from which

The stresses follow from Eqs (ii)

l2Pxy P x a,=---

b3 ^- I Y

u,, = 0 (iii)

where I = b3/12 the second moment of area of the beam cross-section.

subject to the following conditions.

shear stress T~~ given by Eqs (iii).

as those given by Eqs (iii).

We note from the discussion of Section 2.4 that Eqs (iii) represent an exact solution ( 1 ) That the shear force Pis distributed over the free end in the same manner as the (2) That the distribution of shear and direct stresses at the built-in end is the same (3) That all sections of the beam, including the built-in end, are free to distort.

In practical cases none of these conditions is satisfied, but by virtue of St. Venant's principle we may assume that the solution is exact for regions of the beam away from the built-in end and the applied load. For many solid sections the inaccuracies in these regions are small. However, for thin-walled structures, with which we are primarily concerned, significant changes occur and we shall consider the effects of axial constraint on this type of structure in Chapter 11.

We now proceed to determine the displacements corresponding to the stress system of Eqs (iii). Applying the strain-displacement and stress-strain relationships, Eqs (1.27), (1.28) and (1.47), we have

du

_ -

^{ux --} Pxy

E, = - = -

d x E EI

dv

vux vPxy

du dv ryy P

E = - = - - = - d y E EI (v)

= - + - = - = - - ( b ' - 4 8 IG Y 2 ) (vi)

xy d y d x G

Integrating Eqs (iv) and (v) and noting that E, and sY are partial derivatives of the displacements, we find

Px2y uPxy2

u = - -

+fib),

^v=- 2EI + f 2 ( x ) (vii)

where

fi

( y ) andfi(x) are unknown functions of x and y . Substituting these values of

u and ^Y in Eq. (vi)

2EI

px2 ' f i ( Y ) vpY2

ah(x)

^- _{- P (b2 - 4y2)}

+-+-

81G

--

2EI 8y 2EI ax

46 Two-dimensional problems in elasticity

Separating the terms containing x and y in this equation and writing

we have

The term on the r.h.s. of this equation is a constant which means that F1 ( x ) and F2( y ) must be constants, otherwise a variation of either x or y would destroy the equality.

Denoting Fl ( x ) by C and F2( y ) by D gives Pb2 C + D = - - 8IG and

so that

and

Py3 uPy3 h ( Y ) = a - = + D Y + H Therefore from Eqs (vii)

Px2y uPy3 Py3

+

^-

+

^Dy

+

^H

u = - - - -

2EI 6EI 6IG uPxy2 Px3 v = - +-+Cx+F

2EI 6EI

(viii)

The constants C , D, F and H are now determined from Eq. (viii) and the displacement boundary conditions imposed by the support. system. Assuming that the support prevents movement of the point K in the beam cross-section at the built-in end then u = v = 0 at x ⁼ I , y = 0 and from Eqs (ix) and (x)

H = 0 , F = - - - C l pi3 6EI

If we now assume that the slope of the neutral plane is zero at the built-in end then

&/ax = 0 at x = I , y = 0 and from Eq. (x) C = - - PI2

2EI It follows immediately that

2.6 Bending of an end-loaded cantilever 47

and, from Eq. (viii)

PI2 Pb2 2EI 8IG D = - - -

Substitution for the constants

C,

D , F and H in Eqs (ix) and (x) now produces the equations for the components of displacement at any point in the beam. Thus

(xi)

u = - - - -

vPxy2 Px3 P12x PI3 v = - _2EI +--- _{6EI 2EI}

+-

_3EI

The deflection curve for the neutral plane is

px3

PI^^

pi3

b ) y = O

=E-=+=

(xii)

(xiii) from which the tip deflection ( x = 0) is P13/3EI. This value is that predicted by simple beam theory (Section 9.1) and does not include the contribution to deflection of the shear strain. This was eliminated when we assumed that the slope of the neutral plane at the built-in end was zero. A more detailed examination of this effect is instructive.

The shear strain at any point in the beam is given by Eq. (vi) yxy = - - P (b2 - 4y2)

8ZG

and is obviously independent of x . Therefore at all points on the neutral plane the shear strain is constant and equal to

y = - - Pb2

xy 8IG

which amounts to a rotation of the neutral plane as shown in Fig. 2.6. The deflection of the neutral plane due to this shear strain at any section of the beam is therefore equal to

Pb2

- ( I - X ) 8IG

Fig. 2.6 Rotation of neutral plane due to shear in end-loaded cantilever.

48 Two-dimensional problems in elasticity

P b2/8 I G

-.

r/r

^~

-;g--.-.

^I

L - - - - .- - -

(a) ( b )

Fig. 2.7 (a) Distortion of cross-section due to shear; (b) effect on distortion of rotation due to shear.

and Eq. (xiii) may be rewritten to include the effect of shear as Px3 P12x PI3 Pb2

( z l ) y = o =

6EI

^-

E +

^- _{3EI 8IG}

+

^{- (1 - x) (xiv)}

Let us now examine the distorted shape of the beam section which the analysis assumes is free to take place. At the built-in end when x = 1 the displacement of any point is, from Eq. (xi)

vPy3 +--- Py3 Pb2y 6EI 61G 8IG

u=-

The cross-section would therefore, if allowed, take the shape of the shallow reversed S shown in Fig. 2.7(a). We have not included in Eq. (xv) the previously discussed effect of rotation of the neutral plane caused by shear. However, this merely rotates the beam section as indicated in Fig. 2.7(b).

The distortion of the cross-section is produced by the variation of shear stress over the depth of the beam. Thus the basic assumption of simple beam theory that plane sections remain plane is not valid when shear loads are present, although for long, slender beams bending stresses are much greater than shear stresses and the effect may be ignored.

It will be observed from Fig. 2.7 that an additional direct stress system will be imposed on the beam at the support where the section is constrained to remain plane. For most engineering structures this effect is small but, as mentioned previously, may be significant in thin-walled sections.

1 Timoshenko, S. and Goodier, J. N., Theory of Elasticity, 2nd edition, McGraw-Hill Book Company, New York, 1951.

P.2.1 A metal plate has rectangular axes Ox, Oy marked on its surface. The point 0 and the direction of Ox are fixed in space and the plate is subjected to the following uniform stresses:

Problems 49 compressive, 3p, parallel to

Ox,

tensile, 2p, parallel to Oy,

shearing, 4p, in planes parallel to O x and Oy in a sense tending to decrease the angle xOy

Determine the direction in which a certain point on the plate will be displaced; the coordinates of the point are ( 2 , 3 ) before straining. Poisson~s ratio is 0.25.

A m . 19.73" to O x

P.2.2 What do you understand by an Airy stress function in two dimensions? A beam of length 1, with a thin rectangular cross-section, is built-in at the end x = 0 and loaded at the tip by a vertical force P (Fig. P.2.2). Show that the stress distribution, as calculated by simple beam theory, can be represented by the expression

q5 = Ay3

+

By3x

+ ^{C ~ X}

as an Airy stress function and determine the coefficients A , B, C . Ans. A = 2 P l / t d 3 , B ⁼ -2P/td3, C = 3P/2td

Y I

Fig. P.2.2

P.2.3 A thin rectangular plate of unit thickness (Fig. P.2.3) is loaded along the edge y = +d by a linearly varying distributed load of intensity MJ = p x with

Fig. P.2.3

50 Two-dimensional problems in elasticity

corresponding equilibrating shears along the vertical edges at x = 0 and 1. As a solution to the stress analysis problem an Airy stress function q5 is proposed, where

+=--

^{[5(x3 - Z2x)(y}

+

d ) 2 ( y - 2d) - 3yx(y2 - d2)2]

120d3

Show that q5 satisfies the internal compatibility conditions and obtain the distribution of stresses within the plate. Determine also the extent to which the static boundary conditions are satisfied.

P X

ax = - [5y(x2 -

z2)

- ioy3

+

^6dZy]

20d3

ay =

E (

_4d3 ^{- 3yd2 - 2d3)}

-P 40d3

Txy = - [5(3x2 - Z2)(y2 - d 2 ) - 5y4

+

6y2d2 -

$1

The boundary stress function values of ^T~~ do not agree with the assumed constant equilibrating shears at x = 0 and 1.

P.2.4 A two-dimensional isotropic sheet, having a Young’s modulus E and linear coefficient of expansion a, is heated non-uniformly, the temperature being T ( x , y ) . Show that the Airy stress function

4

satisfies the differential equation

V2(V2q5

+

^{E a T ) = 0}

where

is the Laplace operator.