MEC-E8001 Finite Element Analysis, Exam (example) 2018

Transcription

1 MEC-E8 inite Element Analysis Exam (example) 8. ind the transverse displacement wx ( ) of the structure consisting of one beam element and point forces and. he rotations of the endpoints are assumed to be equal in magnitudes but opposite in directions i.e. π Y Y < π. Problem parameters E and I are constants. Zz / / / Xx. Derive the equations of motion of the bar structure shown in terms of axial displacements u X and u X. Use two linear bar elements. External distributed load vanishes and the cross-sectional area A is piecewise constant. EAρ EAρ p Xx. Beam structure of the figure is loaded by force p acting on node. Determine the buckling force p cr of the structure if beam is considered as rigid and displacements are confined to the xz plane. Cross-sectional properties A and I of beam and Young s modulus of the material E are constants. xx zz Yy 4. A thin rectangular slab (assume plane stress conditions) is loaded by a horizontal force and allowed to move horizontally at node whereas nodes and 4 are fixed. Derive the equilibrium equation for the structure according to large displacement theory by using two three-node triangle elements Material parameters C and thickness t at the initial geometry of the slab are constants. 4 Xx 5. A plate strip of width b is simply supported on two edges and free on the other two edges. he plate is assembled at constant temperature Ι. ind the transverse displacement when the upper side temperature is 4Ι and that of the lower side Ι. Assume that temperature does not depend on x or y and use approximation w( x y) < uz( x/ )( x/ ). Problem parameters E ρ and t are constants. Zz Xx

2 ind the transverse displacement wx ( ) of the structure consisting of one beam element and point forces and. he rotations of the endpoints are assumed to be equal in magnitudes but opposite in directions i.e. πy < πy. Problem parameters E and I are constants. Zz / / / Xx Virtual work expression of the internal forces int z 6h 6h uz y EI yy 6h 4h 6h h πy z h 6 6 z y πy χu χπ < χu h h u χπ 6h h 6h 4h is available in the formulae collection. However external part for forces is given only for a constant distributed force and point forces acting on the nodes. Here the point forces are acting inside the element. heir contribution follows from the definition of work but the virtual displacement need to be expressed in terms of the displacement and rotation of nodes by using the cubic approximation for bending. he active degrees of freedom are the rotations which satisfy πy < πy ( ω) ( ω) ( ω) ω π Y x w< < x( ) π ( ωω ) π ω ( ω ) Y Y x < x. χ w ( ) χπy At the points of action of the forces and χ w < ( ) χπy < χπy and 9 χ w < ( ) χπy < χπy. 9 herefore the virtual work expression of the point forces ext 4 < χπy χπy < χπy Virtual work expression of the internal forces simplifies to 6h 6h int χπ Y 6h 4h 6h h EI π Y EI < χπ 4 π 6h 6h < χπy 6h h 6h 4h πy Y Y Principle of virtual work and the fundamental lemma of variation calculus imply.

3 int ext EI 4 < < χπy(4 πy ) < 9 π Y <. 9 EI x w < x( ) 9 EI.

4 Derive the equations of motion of the bar structure shown in terms of axial displacements u X and u X. Use two linear bar elements. External distributed load vanishes and the cross-sectional area A is piecewise constant. Bar element contributions of the formulae collection are EAρ EAρ Xx int x EA ux χu < χu h u x x and ine x θah u%% x χu < χu 6. u%% x x rom the figure the nodal displacement of bar are u x < and ux < ux. herefore EA θa < ( ) χu u 6. u%% X X X he nodal displacement of bar are ux < ux and ux < ux. herefore X EA X θa %% X χu u u < ( ) χu u 6. u%% X X X Virtual work expression of the structure is the sum of element contributions X EA X θa 6 %% X χu u u < < ( ) χu u 6. u%% X X X Principle of virtual work <! χa and the fundamental lemma of variation calculus χa <! χa < imply that EA ux θa 6 u%% X < u 6. u%% X X

5 Beam structure of the figure is loaded by force p acting on node. Determine the buckling force p cr of the structure if beam is considered as rigid and displacements are confined to the xz plane. Cross-sectional properties A and I of beam and Young s modulus of the material E are constants. he normal force N < p can be deduced without any calculations. Since beam is rigid the displacement and rotations of nodes and are related by πy < πy and uz < πy. et us consider π as the active degree of freedom of the structure. Element contribution for beam taking into Y account the beam bending mode and the interaction of the bar and beam bending modes is given by p xx zz EI p < ( ) χπ Y π Y χπ Y π Y Y EI 6 p 6 πy χπ < ( ) χπ π Y Y EI p < χπ ( 8 46) π Y Y Principle of virtual work and the fundamental lemma of variation calculus imply the equation system EI p ( 8 46) πy <. Clearly either rotation or its multiplier needs to vanish. he critical value of the loading parameter p making the solution non-unique is given by the latter option EI p 8 46 < p 4 EI EI <. cr. 85ο

6 Yy A thin rectangular slab (assume plane stress conditions) loaded by a horizontal force is allowed to move horizontally at node and nodes and 4 are fixed. Derive the equilibrium equation for the structure according to large displacement theory by using two three-node triangle elements Material parameters C and thickness t at the initial geometry of the slab are constants. 4 Xx Virtual work density of internal force when modified for large displacement analysis with the same constitutive equation as in the linear case of plane stress is given by χ E xx E xx E xx u x u x u xu x v xv x int tc χwς χ E yy < E yy Eyy < v y v y u yu y v yv y. χe ( )/ xy Exy Exy u y v x u xu y v xv y As all the nodes of element are fixed it is enough to consider element. et us start with the approximations and the corresponding components of the Green-agrange strain. inear shape functions can be deduced from the figure. Only the shape function N < ( x/ y/ ) of node is needed. Displacement components v < w < and x y u < ( ) u X u u X x < u y < < a E xx Eyy < a a E xy and χ E xx χeyy < χa( a ) χe xy When the strain component expression is substituted there virtual work density simplifies to int tc χwς < χ a( a ) ( a a ). ( )/ erm by term calculation int tc tc ( χwς ) < χ a a χa a < ( )/ tc tc ( χw ) χ a a χa a ( )/ int ς < <

7 tc tc ( χw ) χ aa a χa a ( )/ int ς < < tc tc ( χw ) χ aa a χa a ( )/ int ς 4 < <. Virtual work density is the sum of the terms. Virtual work expression of internal forces follows after integration over the element tc a a( a a ). < χ Virtual work expression of the point force follows from the definition of work < χu < χ. X a Virtual work expression of the structure is obtained as sum over the element contributions tc < < χa[ a( a a ) ]. Principle of virtual work and the fundamental lemma of variation calculus imply the equilibrium equation tc a( a a ) <.

8 A plate strip of width b is simply supported on two edges and free on the other two edges. he plate is assembled at constant temperature Ι. ind the transverse displacement when the upper side temperature is 4Ι and that of the lower side Ι. Assume that temperature does not depend on x or y and use approximation w( x y) < uz( x/ )( x/ ). Problem parameters E ρ and t are constants. Zz Xx Assuming that the material coordinate system is chosen so that the plate bending and thin slab modes decouple in the linear analysis the Kirchhoff plate model virtual work densities of internal force and coupling terms are given by χ w xx w xx int χw χ w yy D ς < w yy χw ( )/ xy w xy where t E D < cpl χ w xx E χwς < z Ιdz χ w Χ. yy he coupling term contains an integral of temperature over the thickness of the plate. Approximation to the transverse displacement and its derivatives are x x w( x y) < uz ( ) w u < and w yy < w xy <. xx Z emperature difference and its weighted integral over the thickness (integral of the coupling term) z z z Χ Ι < Ι( z) Ι < ( ) Ι ( )4Ι Ι < Ι t t t z zχ Ιdz < z Ι dz < Ι t t 6 t/ t/. t/ t/ When the approximation to the transverse displacement is substituted there virtual work densities of the internal and the coupling parts simplify to int D χwς < χuz4 u 4 Z and cpl Ι t E χwς < χuz. Virtual work expressions are integrals of the densities over the domain occupied by the plate/element int bd < χw dxdy < χu u b int ς Z4 Z

9 cpl b cpl b E < χw dxdy u ς < χ Z Ι t. Virtual work expression is the sum of the parts int cpl b D E < < χuz [4 u Z Ι t ]. Principle of virtual work χ W <! χ a and the fundamental lemma of variation calculus give D E u Z Ι t < 4 uz < Ι ( ). t