### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Fem Sol&Str CEE 510

UM

GPA 3.76

### View Full Document

## 13

## 0

## Popular in Course

## Popular in Civil and Environmental Engineering

This 6 page Class Notes was uploaded by Karson Dicki on Thursday October 29, 2015. The Class Notes belongs to CEE 510 at University of Michigan taught by Sherif El-Tawil in Fall. Since its upload, it has received 13 views. For similar materials see /class/231668/cee-510-university-of-michigan in Civil and Environmental Engineering at University of Michigan.

## Reviews for Fem Sol&Str

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/29/15

SHELLS Like plates shells are structures in which the thickness is much smaller than the other two dimensions In fact shells can be considered as plates with membrane action ie can carry forces parallel to their middle surface FORCES AND STRESSES IN SHELLS t2 t2 t2 Recall that plate forces are M Iaxzdz My I ayzdz Mxy I r zdz 42 42 42 xy t2 t2 Q 123612 and Qy I ryzdz 42 42 2 2 In add1tlon to these shells can res1st membrane forces N 0612 N y 039 dz 722 722 2 N Ln 139de Stresses are calculated in the same way For example the xstress can be calculated from the moments at any height 2 as follows M z N x 3 t 12 t 039 Stresses in shells can vary signi cantly within small regions and analysts must be wary of this issue METHOD FOR FORMULATING SHELL ELEMENTS Shells can be thin or thick and can be formulated from plate theory by adding membrane effects by degenerating and imposing speci c constraints on solid elements or by using classical shell theory In the first approach plate elements are modi ed by adding relevant DOFS and stiffness components corresponding to them see figure below We sort of did this for the beam element when we added axial force DOFs to it This approach means that any shell would be approximated using at elements which gives acceptable results as the mesh is refined Elements based on classical shell theory have so many problems that they are not frequently used These elements explicitly consider the curved surface of a shell and take this into consideration in the formulation The main source of the problem is that the membrane action Page I interacts unfavorably with bending effects Since membrane stiffness is much higher than bending stiffness this leads to illconditioning and poor performance This is sometime know as membrane locking Degenerated elements have been most successful and are discussed next But first let s remember a few preliminaries Bending Effects Membrane Effects VECTOR PROPERTIES Consider the following figure on the left The vector jk has length L and is defined by the coordinates of j and k The components of the vector are xk x j yk y j zk 2 The vector is inclined wrt to the main axes by angles at 5 y The cosines ofthose angles are known as the direction cosines l m and n because the components of the vector along the main axes are Lcosa Lcos Lcosy xk xj In other words I and so on k Xkgtykgt Zk waBJB XxyAJA y Page 2 For the gure above on the right we can nd the perpendicular to the plane containing two vectors A and B by crossmultiplication In this case u v w AXB xA yA 2A x3 yB 28 The components of AXB are yA B szA xAzBzAxB xAyB yAxB You can get a unit vector in this direction by dividing the vector components by the magnitude of the vector leBl l m UAZB yBZA xAZB ZAxB xAyB yAxBT iAxB The components of the unit vector are the direction cosines of the vector ISOPARAMETRIC SHELL ELEMENTS You may wonder why not use the solid elements shown below to model shells The problem is that as the solid element becomes thinner the membrane stiffness starts to dominate and locking or illconditioning may results By explicitly enforcing constraints to make the solid element behave in a speci c way ie as a shell does we remove the sources of numerical dif culty Surface does not need to be a plane Thickness may vary at all points Shell elements from degenerated solids have 5 DOFs per node 3 translational and two rotational The drilling degree of freedom is not considered and is usually unimportant for structural behavior Page 3 Consider a typical node 139 J z V Bi 3i c0 Vza Z V C31 y 1 k X oci Let V3i be the vector representing the through thickness line joining j and k The direction cosines ofthis line are 13 quot13 and 713 Using the isoparametric formulation we can say that the coordinates of any point inside the element are x xx 131 t y ZN1 y ZN1 m3 1 z z n 31 In the second gure let s de ned two other vectors V and Vzi that are mutually perpendicular and perpendicular to V3 We will assume that the rotational DOFs xi and ii are parallel to these The arbitrary definition of rotations is possible because the isoparametric nature of the formulation will take care of correlating these to the global DOFs ie through transformations Note that it is possible for each node to have different orientations for the rotational DOFs Since vectors V and Vzi are arbitrary we need to define them in a unique manner This can be achieved by cross multiplying a vertical vector by V3i to give Vli ie j X V we are now sure that V is perpendicular to V3i Page 4 We can then de ne the third direction by crossmultiplying V and V3 By this process we can get the direction cosines of V and Vzi We can then get unit vectors along these vectors and can arrange them in the following manner l 111 2 1 m2 m1 quot2 quot11 To get the straindisplacement relationship let s write out the displacement at any point and differentiate Consider the point P on the vector V3i The displacement at P is comprised of the displacement at the node 139 components due to small rotations xi and E For example along the xaxis up u a Q 2121 pl Q11 21h This can be arranged into the following expression u u t a v ZN1 v1 ZN 3M 2 w w I To get strains we need to differentiate the displacement expression 6 100000000 u I u x x161 001000100 W W 7116 6x1 6x9 Z 9161 Z 9161 x However the expression in Equation 2 above is in natural isoparametric coordinates and not in Canesian coordinates If you recall we addressed this problem though the Jacobian We defined the Jacobian in the following context for a 3D case 1 1 1 1 au 1 a 6i 6i 65 6x 6x 1 1 1 1 1 111 677 677 677 677 6 6y 61 1 1 1 1 1 6gquot 6gquot 6gquot 6gquot 62 62 From Equation 1 you can get terms like these and fill out the Jacobian x9 ZN1 ltxxg111312 Page 5 The inverse of the Jacobian relates differentiation wrt Cartesian coordinates to differentiation with respect to natural coordinates ie 71 uX J 0 0 115 11 0 J71 0 39 Y9x9 71 wZ 9x1 0 0 J w m w4 9x1 Where N1 0 0 ngxglzx 2 ngxgln 2 u u u 1 V1 2 W lezlm W W 961 a a 9x5 A 561 A 5161 Combination of the above equations leads to u u 5x V V Z Xlsw Yl9x9 Z9x5 w 2 B1 6gtc5 W 72 61 0 a 1 5161 5161 Now that we have the strains let s consider the stresses in the local coordinate system for a homogenous isotropic system The local system varies from one point to the other and is related to the global system through transformation matrices recall our lectures on transformations 0391 E uE 0 0 0 0 51 0392 uE E 0 0 0 0 5 0393 0 0 0 0 0 0 g 13912 2 0 0 0 G 0 0 yu 13923 0 0 0 0 G 0 ya 13931 0 0 0 0 0 G y Note a few things about the above matrix 1 In the above 123 is the local system 12 is tangent to the mid surface and 3 is perpendicular to 12 2 In the 12 direction the shear stresses are related to the shear strains through a G that is independent of the local coordinates Page 6

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.