3.2 The Stress Function Method

Size: px

Start display at page:

Download "3.2 The Stress Function Method"

Evan Barker
7 years ago
Views:

1 Section.. The Stress Fnction Method An effective wa of dealing with man two dimensional problems is to introdce a new nknown, the Air stress fnction, an idea broght to s b George Air in 186. The stresses are written in terms of this new fnction and a new differential eqation is obtained, one which can be solved more easil than Navier s eqations...1 The Air Stress Fnction The stress components are written in the form (..1) Note that, nlike stress and displacement, the Air stress fnction has no obvios phsical meaning. The reason for writing the stresses in the form..1 is that, provided the bod forces are zero, the eqilibrim eqations are atomaticall satisfied, which can be seen b sbstitting Eqns...1 into Eqns... { Problem 1}. On this point, the bod forces, for eample gravitational forces, are generall ver small compared to the effect of tpical srface forces in elastic materials and ma be safel ignored (see Problem of.1). When bod forces are significant, Eqns...1 can be amended and a soltion obtained sing the Air stress fnction, bt this approach will not be followed here. A nmber of eamples inclding non-zero bod forces are eamined later on, sing a different soltion method... The Biharmonic Eqation The Compatabilit Condition and Stress-Strain Law In the previos section, it was shown how one needs to solve the eqilibrim eqations, the stress-strain constittive law, and the strain-displacement relations, reslting in the differential eqation for displacements, Eqn..1.. An alternative approach is to ignore the displacements and attempt to solve for the stresses and strains onl. In other words, the strain-displacement eqations.1. are ignored. However, if one is solving for the strains bt not the displacements, one mst ensre that the compatibilit eqation 1..1 is satisfied. Eqns...1 alread ensres that the eqilibrim eqations are satisfied, so combine now the two dimensional compatibilit relation and the stress-strain relations.1.1 to get { Problem } 6 Kell

2 Section. plane stress : plane strain : 1 (..) Ths one has what is known as the biharmonic eqation: The biharmonic eqation (..) The biharmonic eqation is often written sing the short-hand notation. B sing the Air stress fnction representation, the problem of determining the stresses in an elastic bod is redced to that of finding a soltion to the biharmonic partial differential eqation.. whose derivatives satisf certain bondar conditions. Note that the biharmonic eqation is independent of elastic constants, Yong s modls E and Poisson s ratio. Ths for bodies in a state of plane stress or plane strain, the stress field is independent of the material properties, provided the bondar conditions are epressed in terms of tractions (stress) 1 ; bondar conditions on displacement will bring the elastic constants in throgh the stress-strain law. Frther, the plane stress and plane strain stress fields are identical... Some Simple Soltions Clearl, an polnomial of degree or less will satisf the biharmonic eqation. Here follow some elementar eamples. (i) (ii) (iii) A one has A,, a state if niaial tension B here,, B, a state of pre shear A B here, A,, B, a sperposition of (i) and (ii) 1 technicall speaking, this is tre onl in simpl connected bodies, i.e. ones withot an holes, since problems involving bodies with holes have an implied displacement condition (see, for eample, Barber (199),.). 7 Kell

3 Section... Pre Bending of a Beam Consider the bending of a rectanglar beam b a moment M, as shown in Fig...1. The elementar beam theor predicts that the stress varies linearl with, Fig...1, with the ais along the beam-centre, so a good place to start wold be to choose, or gess, as a stress fnction C, where C is some constant to be determined. Then 6C,,, and the bondar conditions along the top and bottom of the beam are clearl satisfied. b M M Figre..1: a beam in pre bending The moment and stress distribtion are related throgh M b b d 6C d Cb (..) b b and so C M / b and M / b. The fact that this last epression agrees with the elementar beam theor ( M / I with I b h /, where h is the depth into the page ) shows that that the beam theor is eact in this simple loading case. Assme now plane strain conditions. In that case, there is another non-zero stress component, acting perpendiclar to the page, zz ( ) M / b. Using Eqns..1.1b, 1 E 1 (1 ) E 1 E M, sa b v(1 ) M (1 ), sa E b (..5) and the other for strains are zero. As in 1.., once the strains have been fond, the displacements can be fond b integrating the strain-displacement relations. Ths 8 Kell

4 Section. f ( ) 1 g( ) 1 1 g( ) f ( ) f ( ) g( ) (..6) Therefore f () mst be some constant, C sa, so f ( ) C A, and 1 g( ) C B. Finall, C A 1 1 C B (..7) which are of the form For the case when the mid-point of the beam is fied, so has no translation, (,) (,), and if it has no rotation there, (,), then the three arbitrar constants are zero, and z 1 1 (..8)..5 A Cantilevered Beam Consider now the cantilevered beam shown in Fig.... The beam is sbjected to a niform shear stress over its free end, Fig...a. The bondar conditions are (, ), (, ), (, b) (, b) (..9) It is difficlt, if not impossible, to obtain concise epressions for stress and strain for problems even as simple as this. However, a concise soltion can be obtained b relaing one of the above conditions. To this end, consider the similar problem of Fig...b this beam is sbjected to a shear force F, the resltant of the shear stresses. The applied force of Fig...b is eqivalent to that in Fig...a if b b (, ) d F (..1) an eact soltion will sall reqire an infinite series of terms for the stress and strain 9 Kell

5 Section. This is known as a weak bondar condition, since the stress is not specified in a pointwise sense along the bondar onl the resltant is. However, from Saint-Venant s principle (Part I,..), the stress field in both beams will be the same ecept for in a region close to the applied load. b F ( a) (b) Figre..: A cantilevered beam sbjected to; (a) a niform distribtion of shear stresses along its free end, (b) a shear force along its free end The elementar beam theor predicts a stress M / I F / I. Ths a good place to start is to choose the stress fnction, where is a constant to be determined. The stresses are then 6,, (..11) However, it can be seen that (, ) b b. To offset this, one can sperimpose a constant shear stress b, in other words amend the stress fnction to b (..1) The bondar conditions are now satisfied and, from Eqn...1, and so F b, F (..1) b, F b (..1) b..6 Problems 1. Verif that the relations..1 satisf the eqilibrim eqations.... Derive Eqn..... A large thin plate is sbjected to certain bondar conditions on its thin edges (with its large faces free of stress), leading to the stress fnction 5 Kell

6 Section. 5 A B (i) se the biharmonic eqation to epress A in terms of B (ii) calclate all stress components (iii) calclate all strain components (in terms of B, E, ) (iv) derive an epression for the volmetric strain, in terms of B, E,, and. (v) check that the compatibilit eqation is satisfied (vi) check that the eqilibrim eqations are satisfied. A ver thick component has the same bondar conditions on an given crosssection, leading to the following stress fnction: 5 (i) is this a valid stress fnction, i.e. does it satisf the biharmonic eqation? (ii) calclate all stress components (with 1/ ) (iii) calclate all strain components (iv) find the displacements (v) specif an three displacement components which will render the arbitrar constant displacements of (iv) zero 5. For the cantilevered beam discssed in..5, evalate the resltant shear force and moment on an arbitrar cross-section. Are the as o epect? (Yo will find that the beam is in eqilibrim, as epected, since the eqilibrim eqations have been satisfied.) 6. For the cantilevered beam discssed in..5, evalate the strains and displacements, assming plane stress conditions. Note: to evalate the three arbitrar constants of integration, one wold be tempted to appl the obvios all along the bilt-in end. However, since onl weak bondar conditions were imposed, one cannot enforce these strong conditions (tr it). Instead, appl the following weaker conditions: (i) the displacement at the bilt-in end at is zero ( ), (ii) the slope there, /, is zero. 7. Show that the stress fnction p 5 L 15h h 5h h satisfies the bondar conditions for the simpl spported beam sbjected to a niform pressre p shown below. Check the bondar conditions in the weak (Saint- Venant) sense on the shorter left and right hand sides (for both normal and shear stress). Since the normal stress is not zero at the ends, bt onl its resltant, check also that the moment is zero at each end. p Lp Lp h L L 51 Kell

7 Section. Note that the elementar beam theor predicts an approimate fleral stress bt an eact shear stress: 6 p 6 p h L, h h 8. Consider the dam shown in the figre below. Assme first a general cbic stress fnction C1 C C C 6 6 Appl the bondar conditions to determine the constants and hence the stresses in the dam, in terms of, the densit of water. (Use the stress transformation eqations for the sloped bondar and ignore the weight of the dam.) [Jst consider the effect of the water; to these mst be added the stresses reslting from the weight of the dam itself, which are given b 1, s g, tan where is the densit of the dam material.] s 5 Kell

Introduction to Plates

Chapter Introduction to Plates Plate is a flat surface having considerabl large dimensions as compared to its thickness. Common eamples of plates in civil engineering are. Slab in a building.. Base slab