Beams - University of Nebraska–Lincoln. Beams.pdf · 2017-11-28 · 1) Derive the member local stiffness values for two-dimensional beam members. 2) Assemble the local stiffness

Advanced Structural Analysis

Beams © Richard L Wood, 2017 Page 1 of 45

Beams

Lesson Objectives:

1) Derive the member local stiffness values for two-dimensional beam members.

2) Assemble the local stiffness matrix into global coordinates.

3) Assemble the structural stiffness matrix using direct stiffness, applied unit displacements,

and code numbering techniques.

4) Outline procedure and compute the response of beams using the stiffness method.

Background Reading:

1) Read Kassimali – Chapter 5 .

Introduction:

1) What is a beam?

a. A long straight member.

b. Externally loaded where the forces and couples acting on it are on the plane of

symmetry of its cross section.

i. This includes the reactions.

c. Subjected to ___________________________ and __________ (predominately).

i. Neglect any ____________________ effects.

2) Let’s first detail an idealized model, before discussion on the derivation of the member

stiffness matrix for beam. Similar procedure to that of truss elements.

Analytical Model:

1) A continuous beam can be modeled as a series of _______________________________.

a. Subdivide a member if necessary to have a constant ________________________

along the member’s length.

2) Members are connected at each ends to _______________________________________.



3) The unknown external reactions only act at the _________________________.

4) The joints are modeled as _____________________________.

a. That is, the corresponding ends of the adjacent members are _________________

_____________________________ to the joints.

b. Satisfy the ________________________ and _____________________

conditions for the actual structure.

c. If a certain degree of freedom needs to be eliminated, example for a hinge, then

_____________________ are used as needed. (more to come on this later…)

5) See the example illustrated within Figures 1-3.

Figure 1. Physical diagram of beam1.

Coordinate Systems:

1) The Global Coordinate System is oriented as:

a. Global coordinate system is a right-handed XYZ coordinate system.

a. The X axis is oriented in the horizontal direction, where the positive direction is

defined to the ____________ and coincides with the _______________________

__________________ of the beam in the undeformed state (horizontal).

b. The Y axis is oriented in the vertical direction, where the positive direction is

____________________________.

c. All external loads lie in the XY plane.

1 All figures in Beams were modified from: Kassimali, Aslam. (2012). Matrix Analysis of Structures. 2nd edition. Cengage Learning.



2) Origin is typically placed at the leftmost joint of the beam (see example).

a. Therefore all joints have ____________________________________.

3) The Local Coordinate System is oriented as:

a. Local coordinate system is a right-handed XYZ coordinate system.

b. The X axis is typically oriented in the horizontal direction, where the positive

direction is defined to the right and coincides with the centroidal axis of the beam

in the undeformed state.

c. The Y axis is typically oriented in the vertical direction, where the positive

direction is upwards.

4) Transformation from the Local System to Global System.

a. If they coincide where local x and y is orientated to the positive directions are the

same as the global X and Y, respectively: _______________________________.

b. This only holds true for a ____________________________ member, where the

transformation as shown previously is: ________________________.

5) Refer below to Figure 2.

Figure 2. Discretized beam model identifying the two coordinate systems.



Degrees of Freedom (DOFs):

1) Identification of DOFs are essential for accurate structural analysis.

2) What are degrees of freedom:

a. Independent joint deformation (translation, rotation) that are required to

characterize the deformed shape of the structe under arbitrary loading.

b. Also known as the degree of kinematic indeterminacy of a structure.

3) Beam structures: a joint can have up to two DOFs.

a. __________________________________

b. __________________________________

4) General formula appears as:

# # #

5) For a beam structure, the formula simplifies to:

6) Refer below to Figure 3.

Figure 3. Discretized model identifying the degrees of freedom.



Degrees of Freedom (cont’d):

1) The text (chapters 3 and 5) specificies an approach for identifying and numbering DOFs

for a considered structure.

a. Consistency leads to organization.

b. Identity directly on numerical model using number arrows, where:

i. Restraint are with an arrow and a slash.

ii. This same approach is used for applied forces and reactions.

c. Number starts at the lowest numbered joint and proceeds to the highest numbers,

where

i. Free DOFS are identified first.

ii. Identify translational DOFs before rotational DOFs.

1. Identify x-direction before y-direction, if more than one DOF

exists at a joint.

a. For a horizontal beam, there is no x-direction.

iii. Upon completion of free DOFs labeling, continue routine for numbering

restrained DOFs.

2) Resultant of DOF numbering is a joint displacement vector, _______.

Member Level Stiffness Relationship:

1) To develop the member stiffness relationships, let’s derive it for an arbitrary member m.

2) When member m is subjected to external loads:

a. The member will _________________.

b. __________________________________and _________________are induced at

the member ends.

3) Refer to Figure 4.



Figure 4. Example beam shown with external loading, note the location of member m.

4) The member has ___________ DOFs.

a. Each _________________is required to specify the displacement of m member

ends under arbitrary loads.

b. Member forces and displacements are defined in the _______________________

_________________________________________.

c. The DOFs are numbered in the same manner as done previously (see truss notes).

Figure 5. Member m shown with the member end forces and displacements in local coordinates.



5) The relationship between the __________________________________________ and the

____________________________________ can be determined by subjecting the member

to each DOF separately (keeping the other DOFS restrained) as well as applied fixed end

forces due to external loading.

6) Then the member end forces can be expressed using an algebraic sum:

7) kij is a __________________________________________________________________.

8) Qfi are known as __________________________________________________________.

a. These represent the forces that would develop at the member ends due to the applied

external loads, if both ends of the member are fixed.

9) Rewriting the system of equations in matrix form:



Figure 6. Derivation of stiffness coefficients by imposed unit displacements: ________.








Figure 10. Relationship of member end forces due to external loads applied along the member.

Deriving the Member Stiffness Coefficients:

1) Various methods can be used to derive the stiffness coefficients in terms of E, I, and L.

a. Including the consistent deformations (flexibility) or slope-deflection methods.

2) Can derive the coefficients using direct integration of the differential equation for beam

deflection.

a. Will determine ___________________________ and the member shape functions.

b. A shape function represents the _________________________________ along the

____________________ of a member. This is also known as a ____________

______________________________________.

3) Recall from mechanics of materials that:

4) This differential equation is valid for:



a. _________________________________________________________________.

b. _________________________________________________________________.

c. __________________________plane of symmetry of its cross section.

5) In the above equation, the terms as defined as:

a. is defined as the __________________________________beam’s centroidal

axis (neutral axis) in the y-direction at distance x from the origin.

b. denotes the __________________________________at distance x from the

origin.

c. is considered positive in accordance to the following beam sign convention:

_________________________________________________________________.



First column of the member stiffness matrix:

1) To determine the stiffness coefficients due to the _________ degree of freedom, impose a

unit valued displacement at _______ at beam end _______.

a. All other displacements are restrained.

b. Write an equation for the bending moment:

c. Now substitute the equation for the moment and performing some integration:





Second column of the member stiffness matrix:










Third column of the member stiffness matrix:










Fourth column of the member stiffness matrix:










Assembly of the Member Stiffness Matrix:

1) By substitution of previous derived coefficients, the local stiffness matrix for a beam

member can be expressed as:

Transformation from Global to Local Coordinates:

1) Since the beam member local coordinates are typically aligned with the global axes for

the structure, no transformation is often necessary.

2) Beam global stiffness matrix:

3) Relationship for member end forces:

Member Fixed-End Forces Due to External Loads:

1) Stiffness relationship as previously derived now contains additional terms: ________

_____________________________.

a. Forces that develop at the member ends as a result of loading along the _____

____________________.



2) How to obtain fixed-end force expressions?

a. Many different static indeterminate procedures exist.

3) Resultant vector arranged in a similar fashion as the DOFs.

4) The sign convention used must be followed as previously defined.

a. Fixed-end shears are considered as positive when ______________________.

b. Fixed-end moments are considered positive when ______________________.

Figure 11. Example fixed-end force expressions.

5) Therefore if the force shown in Figure 11 was applied to a member, the fixed-end force vector would be:



Structure Stiffness Relations:

1) Now it is desirable to develop the stiffness relationship for the entire structure.

2) This procedure is nearly identical to that of plane trusses.

3) Recall the relationship as:

4) The first method to obtain the structure stiffness relationship uses the _________________

_________________________________________.

a. Equilibrium at joints: Express _____________________ in terms of ___________

________________________________.

b. Compatibility at joints and member ends: Relate ___________________________

to _____________________________________.

c. Member stiffness relationships: joint displacements {d} to the joint loads {P} using

the member-force relation, ________________________________.

5) To outline the procedure, let’s consider an example.

a. A continuous beam structure is identified below in the Figure 12.

b. This beam has ______ degrees of freedom, identified as: ____________________.

c. The discretized beam is illustrated in Figure 13.



Figure 12. Example continuous beam structure subjected to arbitrary loading.

Figure 13. Discretization of example beam.



(a) (b)

(c) (d)

(e)

Figure 14. Details on the member end forces (a, c, and e) and joints (b and d).



Direct Stiffness Method:

1) Equilibrium: apply available equilibrium equations at each node where a DOF is present.

a. By examination of node 2, one can write:

b. By examination of node 3, one can write:

2) Compatibility: inspect the boundary conditions, member ends, and joint conditions.

a. By examination of member 1, one can write:

b. By examination of member 2, one can write:

c. By examination of member 3, one can write:

3) Member stiffness relationships: link the aforementioned relationships of equilibrium and

compatibility. One can express the member stiffness relationship as:



a. Where in expanded matrix form, this is written for member 1 as:

b. The member end forces ______ and _______ for member 1 can be expressed as:

c. The member end forces ______ , ______ and _______ for member 2 can be

expressed as:

d. The member end force ______ for member 3 can be expressed as:

e. Now through the substitution of the compatibility equations into the member

stiffness relationships, one can express:



4) Finally through a substitution into the joint equilibrium equations in step 6, one can write

the desired structure stiffness relationships as:

a. Which can be expressed in matrix form as:

5) Where the structure stiffness matrix is:

6) Where the structure fixed-end force vector is:

7) The preceding approach of combining the member stiffness relations is commonly termed

the ___________________________________________________.

8) [S] is termed the structure stiffness matrix, which is a square matrix of order equal to

__________________.



a. [S] is ________________________________ always for linear-elastic structures.

Imposed Unit Displacement Method:

1) [S] can be also be determined by imposing unit valued displacements for each DOF.

a. The example continuous beam is subjected to unit values of for each DOF (joint

displacements). These are illustrated in Figures 15-17.

(a)

(b)

Figure 15. Example continuous beam structure subject to _____________.



2) By examining Figure 4, a unit displacement is induced at _________, while restraining

_________ and _________.

a. Releasing this DOF, induces member end displacements of _________ and

_________.

b. Member stiffness coefficients develop as a result of aforementioned member end

displacements, shown in part Figure 4b.

i. The explicit values of these stiffness coefficients were previously derived,

as a function of ____, ____, and ____.

c. Therefore by evaluating the resultant stiffness coefficients, the total vertical force

required to create the unit displacement is algebraically summed.

i. Writing the expression for the first coefficient:

ii. Writing the expression for the second coefficient:

iii. Writing the expression for the third coefficient:

d. The three structure stiffness coefficients above represent the first _____________

of the structure stiffness matrix.



(a)

(b)





_________ and _________.

a. Releasing this DOF, induces member end displacements of ______ and ________

along with the corresponding member stiffness coefficients in Figure 5b.

b. As previously done, expressing the total force required to create the unit

displacement is algebraically summed:

c. The second set of three structure stiffness coefficients represent the second

____________ of the structure stiffness matrix.


_________ and _________.

a. Releasing this DOF, induces member end displacements of ______ and ________

along with the corresponding member stiffness coefficients in Figure 6b.

b. As previously done, expressing the total force required to create the unit

displacement is algebraically summed:

c. The third set of three structure stiffness coefficients represent the third

____________ of the structure stiffness matrix.

5) The foregoing stiffness coefficients (_____) are identical to those derived previously.



(a)

(b)


Structure Stiffness Matrix and Structure Fixed Joint Forces using Code Numbers:

1) Formulation of ________, ________, and ________ can be performed by using a

prescribed numbering system to correctly position and assemble the elements.

2) This method of directly formulating the structure stiffness method by code numbers can

be programmed with ease on computer platforms.

3) To illustrate the procedure, let’s examine the previous continuous beam structure

example (Figure 18).



Figure 18. Discretization of example continuous beam.

4) When establishing the code numbers for a member stiffness, the ____________________

of the member will dictate the end joints.

a. Otherwise known as the location of node i and node j.

5) By applying the compatibility equations to member 1, the code numbers correspond to:

_________________________. Therefore this implies:







8) To find the structure stiffness matrix, the individual member stiffness matrices are

aligned. This “matching” procedure can be illustrated below:

9) After partitioning the matrices, an algebraic sum of the relevant elements in their aligned position will produce the structure stiffness matrix.



10) A similar procedure can be carried out to find the structure fixed-joint forces and

equivalent joint loads.

a. The formulation of the ____________ can be performed by assembling the

elements of the member _____________________________ using code numbers.

b. The structure fixed-joint forces represent the _____________________________

that would developed at locations and in the direction of the structure degrees of

freedom, due to an external member load, if all joints of the structure were

___________________.

i. This specifically includes ______________________________________.

c. Acknowledging that if the structure was fully restrained, the forces developed at

the direction and location of the degrees of freedom are ____________________.

d. Using the code numbers as outlined previously identified, the member end force

vector can be indexed with code numbers to find ________.

e. A similar approach can be used to find _____________.



Equivalent Joint Loads:

1) Many tables index equivalent joint loads (or equivalent nodal loads).

a. These are the negative of the ______________________________.

b. Equivalent joint loads are represented by forces and moments at the member ends

to ‘relocate’ the member forces to joint loads.

2) The relationship between equivalent joint loads and structure fixed joint forces is:

3) Therefore when using tables, be mindful of the orientation of the forces!

General Procedure for Analysis for Beams:

The procedure for beams is very similar to that of plane trusses. The general steps can be

summarized as:

1) Discretize an analytical model to represent the structure.

a. Draw and label line diagram.

b. Establish the local and global coordinate systems.

c. Identity the degrees of freedom and the restrained coordinates.

2) Construct the structure stiffness matrix, ______, and fixed-joint force vector, _________.

a. Determine each member’s global stiffness matrix, ______.

i. Determine local stiffness matrix, ______.

ii. Apply the transformation matrix, ______, _________________________.

b. Evaluate the numerical values of ______.

c. Identify the code numbers and assemble the elements of ______ and ______.



d. Note the assembly of ______ and ______ can be found using alternative methods.

i. ___________________________________________________________.

ii. ___________________________________________________________.

3) Formulate the joint load vector, ______.

4) Determine the joint displacements (DOFs) using the structure stiffness relationship:

a. Joint translations are considered positive when ___________________________.

b. Joint rotations are considered positive when _____________________________.

5) Use the computed joint displacements to find the member end forces, member end

displacements, and support reactions as needed.

6) Check equilibrium to verify the solution.



Beams: Examples

Example #1

For the concrete beam illustrated below, determine the joint displacements, member end forces, and the support reactions. Use the matrix stiffness method.

Additional Information:Nodes 1 and 3 are fixed

P = 25 kip M = 60 kip-feet

L1 = 15 feet L2 = 15 feet E = 4500 ksi I = 600 in4

Solution:

Identify the number of degrees of freedom. NDOF = ________.

Identify the number of reactions. NR = ________.

E = _____________.

I = _____________.



Finding the stiffness matrix for member 1.

L =


L =



Assembling the Structure Stiffness Matrix.

Assemble the Joint Load Vector.

Find the Joint Displacements.



Compute Member End Displacements and Forces.

Compute the Reactions.



Example #2

For the steel beam illustrated below, determine the joint displacements, member end forces, and the support reactions. Use the matrix stiffness method.

Additional Information:

Node 1 is fixed Node 2 is an idealized roller

P = 85 kN L1 = 8 meters L2 = 4 meters E = 200 GPa

I = 700(106) mm4:

Solution:

Identify the number of degrees of freedom. NDOF = ________.

Identify the number of reactions. NR = ________.

E = _____________.

I = _____________.




L =


L =



Assembling the Structure Stiffness Matrix.

Assembling the Joint Load Vector.

Find the Joint Displacements.



Compute Member End Displacements and Forces.

Compute the Reactions.

Documents

Beams - University of Nebraska–Lincoln. Beams.pdf · 2017-11-28 · 1) Derive the member local stiffness values for two-dimensional beam members. 2) Assemble the local stiffness