Period #5: Strain. A. Context. Structural mechanics deals with the forces placed upon mechanical systems and the resulting deformations of the system.

Transcription

1 Period #5: Strain A. Contet Strctral mechanics deals with the forces placed pon mechanical sstems and the reslting deformations of the sstem. Solid mechanics on the smaller scale relates stresses in the material to the strains in the materials that comprise the strctre. (a) In this period we ll introdce the concept of strain and the strain tensor. (b) Fig. 5.. (a) nloaded ndeformed strctre and (b) loaded strctre with visible deformation. F 57:9 Intro Mech. Def. Bodies The Universit of Iowa 5.

2 B. Normal Strain Normal strain measres the change in length per nit length of an infinitesimall small fiber of material. Consider the deformation of a rbber ball that is being compressed as shown. If we track the infinitesimal element da of dimensions d o b d it maps to the deformed infinitesimal element da of dimensions d b d in the deformed bod. d d da F da d d F (a) (b) Fig. 5.. (a) ndeformed sphere and (b) deformed sphere. The fiber d deforms to d and its change in length per nit length is: d d d : the normal strain in the -direction Similarl the normal strain in the -direction is: d d d The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.

3 If If If shortening of the material no change in length lengthening of the material While normal strain was defined above over infinitesimal ones the concept can be etended to longer and larger members as well. For eample in Fig. 5. below the average aial strain in a strctral member of initial length L o is =DL/L L L DL DL L o average aial strain Fig. 5.. Etension of a strctral member leading to average aial strain. F C. Shear Strains Shear strain represents the redction in angle (radians) dring deformation between two infinitesimal fibers that were initiall perpendiclar. C.C. Swan The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.3

4 da d d da d d (a) (b) Fig (a) ndeformed sstem and (b) deformed sstem. When the infinitesimal element da eperiences shear deformation the two fibers d and d that were originall orthogonal lose that orthogonalit when the deform to d and d. d d shear strain : d ndeformed d deformed The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.4

5 The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.5 D. The Strain Tensor In solid mechanics the wa to describe how a bod moves and deforms in response to applied loads is throgh a vector displacement field. For each point in a bod defined b initial coordinates () there is a displacement vector: When neighboring points in the bod move differentl this leads to deformation or distortion of the bod. To describe the state of deformation at a point in the bod a second rank strain tensor is sed: Note that vales are sed to represent engineering shear strains which have magnitdes twice those of the tensor shear strains. e e e ) ( ) ( ) ( ) ( ε

6 d B d d d d d d A d shear strain : ndeformed deformed tan A A d d d tan B B d A B A B The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.6

7 E. Eample Problems Eample 5. The rigid beam is spported b a pin at A and wires BD and CE. If the allowable normal strain in each wire is ma =. determine the maimm vertical displacement of the load P. The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.7

8 Eample 5. The g wire AB of a bilding frame is originall nstretched. De to an earthqake the two colmns of the frame tilt θ=. Determine the approimate normal strain in the wire when the frame is in this position. Assme the colmns are rigid and rotate abot their lower spports. The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.8

9 Eample 5.3. The block is deformed into the position shown b the dashed lines. Determine: a) the average normal strain along line AB and the shear strain The Universit of Iowa 57:9 Intro Mech. Def. Bodies 5.9

10