definition nextreturn i. beams 1. definition a beam is a long, slender structural member generally...

Definition

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I. Beams

1. Definition A beam is a long, slender structural member generally subjected to transverse loading that produces significant bending effects (transverse displacement rotation).

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Simple beam theory 1

)(

02

)(

0

)(0)()(

0

2

2 xwdxMd

VdxdMordxwdxVdxMdMM

Mmepuilibriumoment

xwdxdVordxxwdVVV

FmepuilibriuForce

A

y

2. Simple beam theory. Assumptions: (A)member cross section is constant.

(B)cross section dimension < overall length. (L/ t>10)

(C) linear variation of stress and strain.(Small deformation theory)

)/(:)(

:

)(:

:

:

lengthforceloadingddistributexw

momentbendingM

transverseforceShearV

rotationorslape

ntdisplacemelateralv

Equilibrium equation of a differential element of the beam.

Beam under distributed load

Differential beam element


Simple beam theory 2

For constant EI and only nodal forces and moments, equation becomes

Solution of displacement v(x) is of cubic polynomial function of x

)()( 2

2

2

2

2

2

xwdxvdEI

dxd

dxMd

04

4

dx

vdEI

2

21

dx

vd

EI

Mk

dx

dv Curvature of the beam for small slope is given by k

: radius of deflected curve.

E: modulus of elasticity.

I: principal moment of inertia about Z-axis

43

2

2

3

1)( axaxaxaxv


Beam element stiffness formulation 1

3. Beam element stiffness formulation. (Direct equilibrium approach) (1) Beam element (No axial effects are considered.)

momentbendingM

forceshearF

rotationnodal

ntsdisplacemenodalvv

ji

ji

:

:

:,

:,

Sign convection of the beam element:

directionypositivepositivvF

ckwisecounterclopositiveM

,:,

,:,

i j

jMiM i j

jj vF ,ii vF ,

x

vy,

Simple beam theory sign convection for positive shear forces and moments.



(2) Assume displacement function (Without distributing loading, w(x)=0)

432

23

1)( axaxaxaxv

Express (x) as a function of nodal displacements as follows.

322

1

432

23

1

3

4

23)(

)(

)0(

)0(

aLaLajdx

Ldv

aLaLaLavLv

adx

dv

avv

j

i

i

),,,(..4)( jjii vvFsODtotalA )0(..)( 4

4

dxvdEIEDbeambasicsatisfyB

.&)( nodesatcontinuityslopentdisplacemeofconditionsthesatistyC

iijijijiji vxxL

vvL

xL

vvL

xv

2

23

23 )(1)(3)(1)(2)(



j

j

i

i

v

v

NNNNdNv

4321 ,,,

.:,,, 4321 elementbeamaforfunctionsshapeNNNN

223

3423

33

322332

32331

1,321

21,321

xLLxL

NLxxL

N

xLLxLxL

NLLxxL

N

In matrix form, we have



(3)Element stiffness Matrix and Equations

Relationships between moment, force and displacement from elementary beam theory are

Using the nodal and beam theory sign convections for shear forces and bending moments, we obtain

2

2

dx

vdEIM

2

2

3

3

2

2

3

3

)(

)(

)0(

)0(

dxLvd

EI

dxLvd

EI

dxvd

EI

dxvd

EI

M

V

M

V

M

F

M

F

f

j

j

i

i

3

3

dx

vdEIV

Beam element Beam theory



Hence, the beam element equation relating nodal forces and nodal displacements is given as

j

j

i

i

j

j

i

i

v

v

LLLL

LL

LLLL

LL

LEI

M

F

M

F

22

22

3

4626

612612

2646

612612

LLLdx

Nd

LLxLxLLxLxLdx

Nd

ddx

Nd

dxvdd

dx

Nd

dxvdd

dxNd

dxdv

dNv

6,12,6,121

)26(),612(),46(),612(1

,,

33

3

2232

2

3

3

3

3

2

2

2

2

Use the following equations into above equation.

dkfor

Where [k] is the element stiffness matrix for a beam element with neglected axial effects.


Example 1

4. Example:

j

j

i

i

M

F

M

F

L

L

LLL

LL

L

EIKK

2

22

3

)2()1(

4

612

264

612612

By direct stiffness method,the system eqn. For the beam is obtained as

EI ConstantP

3

3

2

2

1

1

2

222

22

3

3

3

2

2

1

1

4

612

2644

612661212

00264

00612612

v

v

v

L

L

LLLL

LLL

LLL

LL

L

EI

M

F

M

F

M

F

y

y

y

0,0,:,0: 211332 MMPFloadsvvconditionsBoundary y

0

0

0

4

612

2644

612661212

00264

00612612

0

0

2

1

1

2

222

22

3

3

3

2

v

L

L

LLLL

LLL

LLL

LL

L

EI

M

F

F

P

y

y


Example 1

The final set of equations is

2

1

1

22

223

826

246

6612

0

0

v

LLL

LLL

LL

L

EIP

The transverse displacement at node 1 and rotations at node1 and 2 are

EI

PL

EI

PL

EI

PLv

44

3

12

7 2

2

2

1

3

1

where the minus sign indicates that displacement at node 1 and the positive signs indicate counterclockwise rotations at node1 and 2.

By substituting the known global nodal displacements and rotations into the system equation, we can determine the global nodal forces. The resulting equations are

0

0

4

0

4

312

7

4

612

2644

612661212

00264

00612612

2

2

3

2

222

22

3

3

3

2

2

1

1

EI

PL

EI

PLEI

PL

L

L

LLLL

LLL

LLL

LL

L

EI

M

F

M

F

M

F

y

y

y

The global nodal forces and moments are

PLMPFMPFMPF yy 2

1,

2

3,0,

2

5,0, 332211


Example 2

Local nodal force for each element (used for stress analysis of the entire structure)

2

2

3

2

3

3

3

2

2

)2(

3

3

2

2

2

PL

P

PL

P

y

y

v

v

K

m

f

m

f

Element

PL

P

P

v

v

k

m

f

m

f

y

y

Element

0

2

2

1

1

)1(

2

2

1

1

1

P2

3

P

P

PL PL2

PLP

2

3

Free body diagrams for element 1 and 2 are shown as follows.

According the results of the global nodal forces and moments, the free body diagram for the whole beam is given as shown.

12

3 2

PL

P2

3

P2

5P


Example 3

1 2 3

P2

3

P

M

1 2 3

PL

2

PL

V

Shear force diagram for the beam

Bending moment diagram for the beam

By using the beam theory sign conventions, the shear force V and bending moment M diagrams are shown in the following figures.


Distributed loading

5. Distributed loading Equivalent force system:

Replace the distributed load by concentrated nodal forces and moments tending to have the same effect on the beam as the actual distributing load based on the concept of fixed-end reactions from structural analysis theory.

w

y

1F 2F1M 2M

x

Ex. Distributed load. equivalent force system

w

w

12

2wL12

2wL2

wL

2

wL

30

2wL12

2wL20

3wL

20

7wL

Fixed-end reactions are those reactions at the ends of an element if the ends of the element are assumed to be fixed.

Beam subjected to a distributed load Fixed-end reactions for the beam


Dis EX 1

eM1

eF1

2

wL

12

2wL

Example:

Consider the cantilever beam subjected to the uniform load w. Find the right-end vertical displacement and rotation, and nodal forces.

equivalent nodal force system for uniform load

w

2

2

1

1

2

22

3

2

2

1

1

4

612

264

612612

L

L

LLL

LL

L

EI

M

F

M

F

e

1M

1F

y

w

x 12

2wL 2

wL

2

wL

12

2wL

cantilever beam subjected to the uniform load w

Solution:One element is used to represent the whole beam. Based on the fixed-end reactions concept, the equivalent nodal force system is given as the following figure.

The system equation for the beam is

12

2wL

1M 1F

2

wL

12

2wL

2

wL


Dis EX 1

6

8

12

263

326

3

2

2

2

2

w

wL

EIL

wL

wL

L

LLEIL

Applying the nodal forces and the boundary conditions , we obtain

12,

2

2

22

wLM

wLF 011

2

2

23246

612

12

2v

LL

L

L

EIwL

wL

e

Solving the above equation for the displacements, we have

Therefore, the reaction forces F1e and M1e for the equivalent nodal force system are

125

2

6

826

612

26

612

22

2

2

231

1

wL

wL

w

wL

LL

L

v

LL

L

LEI

M

F

e

e


Dis EX 2

The equivalent nodal forces are

12/

2/

12/

2/

2

2

0

wL

wL

wL

wL

F

Hence, the effective global nodal forces are

EIwL

EIwL

L

L

LLL

LL

LEI

wL

wL

wL

wL

M

F

M

F

F

e

e

e

e

e

6/

8/

0

0

4

612

264

612612

12/

2/

12/5

2/

3

4

2

22

3

2

2

2

2

1

1

Thus, the correct global nodal forces

0

0

2/2

0

wL

wL

FFF e

eMwL

M 1

2

1 12

By comparing the two equivalent system given in the previous page, we have relationships among the correct nodal forces, the effective nodal forces and the equivalent nodal forces.

eFwL

F 11 2


Bar Element 1

DescriptionUniaxial element with tension, compression, torsion, and bending capabilities. The more general Beam element is often used instead of this element. The figure, at the end of this section, defines both element types. For some analysis programs, MSC/N4W translates both types to the same element type.

Application Used to model general beam/frame structures.

Shape Line, connecting two nodes. A third node can be specified to orient the element Y axis.

Element Coordinate System The element X axis goes from the first node to the second.The element Y axis is perpendicular to the element X axis. It points from the first node toward the orientation (or third) node. If you use an orientation vector, the Y axis points from the first node in the direction of the orientation vector. The element Z axis is determined from the cross product of the element X and Y axes.

Properties Area, Moments of Inertia (I1, I2, I12), Torsional Constant, Shear Areas (Y, Z), Nonstructural Mass/Length, StressRecovery Locations.

Additional Notes Refer to the Beam element for further descriptions regarding Releases, Offsets and Stress Recovery Locations.

Bar Element


Bar Element 2

Third Node, or

Orientation Vector

Plane 1 (XY)

Offset A

Offset B

Plane 2 (XY)


Beam Element 1

Description

Uniaxial element with tension, compression, torsion, and bending capabilities. This element can be tapered. You can specify different properties at each end of the beam.

Application

Used to model beam/frame structures.

Shape

Line, connecting two nodes. A third node can be specified to orient the element Y axis.

Element Coordinate System

The element X axis goes from the first node to the second. The element Y axis is perpendicular to the element X axis. It points from the first node toward the orientation (or third) node. If you use an orientation vector, the Y axis points from the first node in the direction of the orientation vector. The element Z axis is determined from the cross product of the element X and Y axes.

Properties

Area, Moments of Inertia (I1, I2, I12), Torsional Constant, Shear Areas (Y, Z), Nonstructural Mass/Length, Stress Recovery Locations, Neutral Axis Offsets (Nay, Naz, Nby and Nbz). If the beam is tapered, you can specify different properties at each end of the element.

Additional Notes

You can specify Releases which remove the connection between selected element degrees of freedom and the nodes.

Offset vectors defined on the Element move the neutral axis and shear center from the nodes. Neutral Axis Offsets (Y,Z) defined on the Property card move the neutral axis away from the shear center. If there are no Neutral Axis Offsets, the neutral axis and shear center are coincident.

Beam Element


Beam Element 2

Third Node, or

Orientation Vector

Plane 1 (XY)

Offset A

Offset B

Plane 2 (XY)

If there are no offsets,both the neutral axis and shear center lie directly between the nodes.

Stress Recovery Locations define positions in the elemental YZ plane (element cross-section) where you want the analysis program to calculate stresses.

Specifying moments of inertia for Beam (and Bar) elements can sometimes be confusing. In MSC/N4W, I1 is the moment of inertia about the elemental Z axis. It resists bending in the outer Y fibers of the beam. It is the moment of inertia in plane 1. Similarly, I2 is the moment of inertia about the elemental Y axis. If you are familiar with one of the analysis program conventions, the following table may help you convert to MSC/N4W's convention.

MSC/N4W I1 I2

MSC/pal & CDA/Sprint Iww Ivv

NASTRAN Izz Iyy

ANSYS IZ1 IY1

STARDYNE I3 I2

ALGOR, mTAB & SAP I3 I2

ABAQUS I22 I11

WECAN Izz Iyy

COSMOS Izz Iyy

STAAD IZ IY


Element coordinate sys.

.:

.,min:

.:

VXproductCrossaxisZElement

planeYXinlieswhichVvectorbyedDeterplaneYXElement

BEndandAEndbetweenlinewithcoincidentAlwaysaxisXElement

e

ee

Element coordinate system. ( orientation of a beam element)

Interpretation of Element Output.

． Bar element internal forces and moments.

(1) plant 1

(2) plant 2

xF

aM1

ey

a b

1VbM1

xF ex

b

ez

aM 2

2Va

bM 2

2V

ex



II. Frames and Grids

1. Rigid plane frame :

(1) Definition

A frame consists of a series of beam elements rigidly connected to each other.

(i) joint angles between elements remain unchanged after deformation

(ii) moment continuity exists at the rigid joint



(iii) element centroids and applied loads lie in the pane of the structure

(2) Two Dimensional Beam element



Vectors transform law in 2D

For the beam element, we have (use the 2nd eq. of above relation)



Global stiffness matrix for 2D Beam element

where

Here, the global stiffness matrix for a beam element including shear and bending effect is given as.



(3) 2D Beam element including axial force effect

Axial force effect

Combined with shear force and bending moment effects



or

or

and relate the local to the global displacement by



The global stiffness matrix for beam element including axial, shear and bending effects

where

The analysis of a rigid plane frame can be performed by using above stiffness matrix.


Grid-1

1F 2F

3F4F

2. Grid (1) Definition A Grid is a structure on which loads are applied perpendicular to the plane of the structure.

(2) Grid Element (Shear + bending + torsion)

x

y

Z

ZZm 11 , xxm 11 , xxm 22 , ZZm 22 ,

11 ,f vy 22 ,f vy


Grid-2

xxm 11 , xxm 22 ,

x x

xxm , xxm ,

Torsional bar element stiffness matrix.

xx L

GJm

where G = shear modulus ; J =centroidal polar moment of inertia

xxxxx mmL

GJm 1212

x

x

x

x

L

GJm

m

2

1

2

1

11

11

Fig. Nodal and element torque sign conventions

The relationship between torque and twist is


Grid-3

We assume the shear loading to go through the shear center of these open cross sections to prevent twisting of the cross section.By Combining the torsional effect with the shear and bending effects, the local stiffness matrix equation for a grid element is written as:

z

x

z

x

b

bb

bbb

bbbb

z

x

y

z

x

y

v

v

kL

k

Lkk

kLLkkL

kk

LkkLkk

m

m

f

m

m

f

2

2

2

1

1

1

2

t

22

tt

2

2

2

1

1

1

4

0

6012

2064

000

60126012

3L

EIkb L

GJkt

dkfor G

Hence, the stiffness matrix for the torsional bar is

11

11

L

GJk

Grid-4

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The transformation matrix relating local to global D.O.F for a grid is

CS

SC

CS

SC

TG

0000

0000

001000

0000

0000

000001

L

xxC ij cos

L

zzS ij sin

Hence, the global stiffness matrix for a grid element is

GG

T

GG TkTk

definition nextreturn i. beams 1. definition a beam is a long, slender structural member generally...

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beam element equation

beam elementbeam theory

elementary beam theory

beam theory sign convections

beam theory sign conventions

element stiffness matrix

differential element

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