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Beam Elements - Matlab

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Beam Elements
Jake BlanchardSpring 2008
Beam Elements
These are “Line Elements,” with2 nodes6 DOF per node (3 translations and 3 rotations)Bending modes are included (along with torsion, tension, and compression)(there also are 2-D beam elements with 3 DOF/node – 2 translations and 1 rotation)More than 1 stress at each point on the element
Shape functions
Axial displacement is linear in xTransverse displacement is cubic in xCoarse mesh is often OKFor example, transverse displacement in problem pictured below is a cubic function of x, so 1 element can give exact solution
F
Beam Elements in ANSYS
BEAM 3 = 2-D elastic beamBEAM 4 = 3-D elastic beamBEAM 23 = 2-D plastic beamBEAM 24 = 3-D thin-walled beamBEAM 44 = 3-D elastic, tapered,unsymmetricbeamBEAM 54 = 2-D elastic, tapered,unsymmetricbeamBEAM 161 = Explicit 3-D beamBEAM 188 = Linear finite strain beamBEAM 189 = 3-D Quadratic finite strain beam
Real Constants
AreaIZZ, IYY, IXXTKZ, TKY (thickness)Theta (orientation about X)ShearZ,ShearY(accounts for shear deflection – important for “stubby” beams)
Shear Deflection Constants
shearZ=actual area/effective area resisting shear
Shear Stresses in Beams
For long, thin beams, we can generally ignore shear effects.To see this for a particular beam, consider a beam of length L which is pinned at both ends and loaded by a force P at the center.
P
L/2
L/2
Accounting for Shear Effects
Key parameter is height to length ratio
Distributed Loads
We can only apply loads to nodes in FE analysesHence, distributed loads must be converted to equivalent nodal loadsWith beams, this can be either force or moment loads
q=force/unit length
M
F
F
M
Determining Equivalent Loads
Goal is to ensure equivalent loads produce same strain energy
Equivalent Loads (continued)
M
F
F
M
Putting Two Elements Together
M
F
F
M
M
F
F
M
M
F
F
F
2F
M
An Example
Consider a beam of length D divided into 4 elementsDistributed load is constantFor each element, L=D/4
qD/8
qD/4
qD/4
qD/8
qD/4
qD2/192
qD2/192
In-Class Problems
Consider a cantilever beamCross-Section is 1 cm wide and 10 cm tallE=100 GPaQ=1000 N/mD=3m, model using surface load and 4 elementsD=3m, directly apply nodal forces evenly distributed – use 4 elementsD=3m, directly apply equivalent forces (loads and moments) – use 4 elementsD=20cm (with and withoutShearZ)
Notes
For adding distributed load, use “Pressure/On Beams”To view stresses, go to “List Results/Element Results/Line elements”ShearZfor rectangle is still 6/5Be sure to fix all DOF at fixed end
Now Try a Frame
F (out of plane)=1 N
3 m
2 m
Cross-sections
6 cm
5 cm

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Beam Elements - Matlab