1. Definition
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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2. 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)
Simple beam theory 1
Beam under distributed load
Equilibrium equation of a differential element of the beam.
Differential beam element

3. Curvature of the beam for small slope is given by
Simple beam theory 2
Curvature of the beam for small slope is given by
r: radius of deflected curve.
E: modulus of elasticity.
I: principal moment of inertia about Z-axis
For constant EI and only nodal forces and moments, equation becomes
Solution of displacement v(x) is of cubic polynomial function of x

4. Beam element stiffness formulation 1
3. Beam element stiffness formulation. (Direct equilibrium approach)
(1) Beam element (No axial effects are considered.)
Sign convection of the beam element:
Simple beam theory sign convection for positive shear forces and moments.

5. Beam element stiffness formulation 2
(2) Assume displacement function (Without distributing loading, w(x)=0)
Express n(x) as a function of nodal displacements as follows.

6. Beam element stiffness formulation 3
In matrix form, we have

7. Beam element stiffness formulation 4
(3)Element stiffness Matrix and Equations
Beam element stiffness formulation 4
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
Beam element
Beam theory

8. Beam element stiffness formulation 5
Use the following equations into above equation.
Hence, the beam element equation relating nodal forces and nodal displacements is given as
Where [k] is the element stiffness matrix for a beam element with neglected axial effects.

9. 4. Example:
Example 1 P
Constant
By direct stiffness method,the system eqn. For the beam is obtained as

10. The final set of equations is
Example 1
The final set of equations is
The transverse displacement at node 1 and rotations at node1 and 2 are
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
The global nodal forces and moments are

11. Free body diagrams for element 1 and 2 are shown as follows.
Example 2
Local nodal force for each element (used for stress analysis of the entire structure)
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. Shear force diagram for the beam
Example 3
By using the beam theory sign conventions, the shear force V and bending moment M diagrams are shown in the following figures.
Shear force diagram for the beam
Bending moment diagram for the beam

13. 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.
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
Ex.
Distributed load.
equivalent force system

14. The system equation for the beam is
Dis EX 1
Example:
Consider the cantilever beam subjected to the uniform load w. Find the right-end vertical displacement and rotation, and nodal forces.
cantilever beam subjected to the uniform load w
equivalent nodal force system for 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

15. Applying the nodal forces and the boundary conditions , we obtain
Dis EX 1
Applying the nodal forces and the boundary conditions , we obtain
Solving the above equation for the displacements, we have
Therefore, the reaction forces F1e and M1e for the equivalent nodal force system are

16. Hence, the effective global nodal forces are
Dis EX 2
Hence, the effective global nodal forces are
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.
The equivalent nodal forces are
Thus, the correct global nodal forces

17. Used to model general beam/frame structures. Shape
Bar Element
Bar Element 1
Description
Uniaxial 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, Stress
Recovery Locations.
Additional Notes
Refer to the Beam element for further descriptions regarding Releases, Offsets and Stress Recovery Locations.

18. Plane 2 (XY) Offset B Plane 1 (XY) Third Node, or Orientation Vector
Bar Element 2
Third Node, or
Orientation Vector
Plane 1 (XY)
Offset A
Offset B
Plane 2 (XY)

19. Used to model beam/frame structures. Shape
Beam Element
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.

20. MSC/pal & CDA/Sprint Iww Ivv NASTRAN Izz Iyy ANSYS IZ1 IY1
Beam Element 2
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 I I2
MSC/pal & CDA/Sprint Iww Ivv
NASTRAN Izz Iyy
ANSYS IZ IY1
STARDYNE I I2
ALGOR, mTAB & SAP I I2
ABAQUS I I11
WECAN Izz Iyy
COSMOS Izz Iyy
STAAD IZ IY
Third Node, or
Orientation Vector
Plane 1 (XY)
Offset A
Offset B
Plane 2 (XY)

21. Element coordinate sys.
Element coordinate system. ( orientation of a beam element)
Interpretation of Element Output.
．Bar element internal forces and moments.
(1) plant 1
(2) plant 2

22. Element coordinate sys.
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

23. Element coordinate sys.
(iii) element centroids and applied loads lie in the pane of the structure
(2) Two Dimensional Beam element

24. Element coordinate sys.
Vectors transform law in 2D
For the beam element, we have (use the 2nd eq. of above relation)

25. Element coordinate sys.
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.

26. Element coordinate sys.
(3) 2D Beam element including axial force effect
Axial force effect
Combined with shear force and bending moment effects

27. Element coordinate sys.
and relate the local to the global displacement by or

28. Element coordinate sys.
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.

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

30. Torsional bar element stiffness matrix.
Grid-2
Torsional bar element stiffness matrix.
Fig. Nodal and element torque sign conventions
The relationship between torque and twist is
where G = shear modulus ; J =centroidal polar moment of inertia

31. Hence, the stiffness matrix for the torsional bar is
Grid-3
Hence, the stiffness matrix for the torsional bar is
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:

32. The transformation matrix relating local to global D.O.F for a grid is
Hence, the global stiffness matrix for a grid element is
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