M Direct Stiffness Method: Plane Frame Example 1Example 111 Degrees of freedom: M = 1. • The bending moment A M2 at end B of member AB. M0
P1
for that element. uT
EId2ydx2
9.
Skew Roller Support BEAM ANALYSIS USING THE STIFFNESS METHOD. y0
The Above equation, can therefore serve as the beam element stiffness matrix for a beam element of length
(5.d), If
Here, meshing is quite minimal as there are only two elements as shown above, and they are enough for this problem. B12 B1B2 B1B3 B1B4B1B2B22B2B3 B2B4B1B3B2B3B32B3B4B1B4B2B4B3B4B42
. M (5.b) ,
x axis (local 1 axis in SAP 2000) i= initial end of element j= terminal end element Note the sign convention Beam Element Stiffness Matrix in Local Coordinates Ke
H.W 4 CE525 THEORY OF MATRIX STRUCTURAL ANALYSIS SUBMITTED BY : KAMARAN SHEKHA ABDULLAH 201568536 DATE : 23 / 11 / 2016 Page 1 L Mi = M j M j Mi +M j L Mi +M j L M M + + M A B M EI M EI M EI + + ML 2EI ML 2EI Mb=0 : (ML/2EI)(L/3)+ (ML/2EI)(2L/3) = 0 M = L/3 L/3 Real Beam Conjugate Beam
are the transverse deflection, rotation, transverse load and moment at x=L
Mi = M j M j is the element stiffness matrix.
For a beam in bending, U b, the potential energy due to bending moment M (x) along the beam is: U b = ∫ 0 L M 2 2 E I d x.
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(K441+K222)
Let us say:
and y1 and
y1=a0+a1L+a2L2+a3L3
M Therefore it defines the shape function.
BT
BT
Equation (5.h) shows that integrating the elements of the right-hand side of equation (5.j) will give the element stiffness matrix for a beam in pure bending: For a beam element with Young's moulus of E,
,
=
The results for displacements are in meters and rotations in radians.
B
SUBMITTED BY : KAMARAN SHEKHA ABDULLAH 201568536 . , which can be written as:
x +a2x2+a3x3
+ In equation (6), y0,
The equation (10) shows how the stiffness matrices of the individual elements are combined to make the global stiffness matrix. . a1=θ0
, or:
θ0,
=
DATE : 23 / 11 / 2016 d
[
M1
For every n×1, i.e. For deriving the element stiffness matrix, total potential energy methods can be used, which is used here. Lecture 12: The Stiffness Method –Prismatic Beams Washkewicz College of Engineering Example 12.1 Consider again the two span beam previously discussed and determine • The shearing force A M1 at end B of member AB. For example, in the slope-deflection method, we ended up with one equation for each degree of freedom.
y0 and
Ke
=
there is no load inside the element. Use the direct stiffness method to solve
θ2
00
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dx
K332
.(4).
Ch 1 structural analysis stiffness method, STIFFNESS MATRIX FOR 3-NODE TRIANGULAR ELEMENT, An Introduction to the Finite Element Method, Análisis de-porticos-por-el-método-de-matriz-de-rigidez, Quick design check of multi storey building, No public clipboards found for this slide, Stiffness matrix method for beam , examples ce525, Student at National Institute of Technology Calicut.