Plane Trusses - University of Nebraska–Lincoln. Plane Trusses.pdfPlane Trusses Notes prepared by: R.L. Wood Page 1 of 40 Plane Trusses Lesson Objectives: 1) Identify the nomenclature

Advanced Structural Analysis

Plane Trusses Notes prepared by: R.L. Wood Page 1 of 40

Plane Trusses

Lesson Objectives:

1) Identify the nomenclature and sign conventions associated with matrix structural

analysis.

2) Derive the member local stiffness values for plane truss members.

3) Assemble the local stiffness matrix into global coordinates using matrix transformations.

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

Background Reading:

1) Read ___________________________________________________________________

Introduction:

1) What is a planar truss?

a. Two-dimensional framework of _______________________________________

connected via _________________________________________ joints (pinned).

b. Externally loaded at ________________________ (and in the plane of the truss).

c. Subjected to ______________________________ only (compression or tension).

2) Let’s focus on some nomenclature before discussion how to create a member stiffness

matrix.

Two Coordinate Systems: Global and Local:

1) Typically two distinct coordinate systems are considered to define a member and global

(or structure) stiffness relationships: __________________and __________________.

2) Global: system where geometry and stiffness (load-deformation link) relationship for the

entire structure are defined.

3) Typically systems are in a Cartesian coordinate system (or _______________________).

a. _________________________________________________________________.

b. Plane structures lie in the XY plane.



c. Origin typically located at the _______________________________ of structure.

4) Local: system defined at the ______________________________ (member-dependent).

a. Define local coordinate systems due to ease of construction of the stiffness

relationships.

i. Locally: ____________________________________ to longitudinal axis.

b. Origin is located at either end (typically referred to as i or j).

c. The x-axis is along undeformed member’s _______________________________.

d. The y-axis is defined using the right handed rule where the z-axis points in the __ _________________________________________________________________.

Figure 1. Example plane truss with external loads1.

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



Figure 2. Example plane truss: defined coordinate systems.



Degrees of Freedom (DOFs):

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

2) What are degrees of freedom or DOFs:

a. Independent _______________________________ (translation, rotation) that are

required to characterize the _________________________of the structure under

arbitrary loading.

b. Also known as the degree of __________________________________________

of a structure.

3) Plane trusses: joints are where DOF are typically located.

a. The frictionless pins do not restrain ____________, therefore the member ends

are not considered as rotational DOF.

b. Therefore, only ____________________________ can occur.

4) Frame structure: number of DOFS equals the number of deformations that can occur

(more on this later).

a. This would include _________________________ and ____________________.

5) General formulae:

6) For plane trusses, the formula simplifies to:



Degrees of Freedom (cont’d):

1) Textbook (chapter 3) specificies an approach for identifying and numbering DOFs for a

considered structure.

a. Identity directly on numerical model using numbered arrows, where:

i. Restraints are shown with an arrow and a slash.

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

b. Number starts at the lowest numbered joint and proceed to the higher numbers,

where

i. _____________________________________ are identified first.

ii. Identify ____________________ DOFs before ________________ DOFs.

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

exists at a joint.

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

DOFs.

2) Resultant of DOF numbering is a joint displacement vector denoted as ____________.

Additional Nomenclature:

1) Joint loads and reactions vectors are considered in a similar procedute as DOF

numbering.

2) Joint Load Vector, denoted as ____________.

a. Considers external loads applied to the joints, resolved into the corresponding __

______________________________________________.

b. Direction and numbering of load vector is consistent with _________________.

3) Reaction Vector, denoted as ____________.

a. Reactions exerted on the structure at a particular location that prevent

______________________________.

b. Reactions are resolved into correspodning x and y components.

c. Direction and numbering of reaction vector is consistent with DOFs.



Figure 3. Example plane truss: displaced shape.



Figure 4. Example plane truss: degrees of freedom, joint load, and reactions.



Member Level Stiffness Relationship:

1) In general, for a structure, the stiffness method determines the joint displaceent ([d]) due

to external loading ([P]), by solving the system of _______________________________:

2) Similar relationship exists for each of the members (or elements) that make up the entire

structure.

3) The member local stiffness expresses forces at each end of the member as a function of

_________________________________________.

4) Let’s first derive the local member stiffness for a truss member.

a. Arbitrary member (m), has length L, area A, and constant modulus E (prismatic).

b. This member m is subjected to _____________________ and _____________ as:

Figure 5. Plane truss subjected to external loads.



Figure 6. Deformed member m in local coordinates.

5) This member m has _____ DOFs.

a. Each DOF is needed to specify the _____________________ of m member ends.

b. Member forces and displacements are defined in the ________ coordinate system.

6) Numbering scheme:

a. Member end displacements and forces are numbered by beginning at the ____ end

of member m, where the origin is located.

b. Translations and forces are numbered first in the ___________________ and then

the translation and force is numbered in the ___________________.

c. At the other end, ___, the translations and forces are numbered in the same order.

7) Need to derive a relationship between the member forces, denoted as _____, and the

member displacements, denoted as ______.

a. Subject member m to each of the four translations _________________________.

b. Compute the total end forces as an ____________________ of end forces

required to cause __________________________________________ (per DOF).

c. Use statics and mechanics of materials to derive ___________________________

_____________________________ for each unit translatieson (per DOF).



8) The total member end force can be found by ___________________________________

required to create each ______________________________________________ (DOFs).

9) Therefore, one can write a system of equations as:

10) In these equations, kij represents the force at the location and in the direction of ____, due

to a ____________________________ displacement at ____, when all other end

displacements are restrained.

a. kij is known as a ___________________________________________________.

11) Rewriting the system of equations in matrix form:

12) Next, let’s determine the values of each cofficient by imposing unit displacements for

each DOF (while keeping the other DOFs restrained)



(a)

(b)

(c)

(d)

Figure 7. Derivation of stiffness coefficients by imposing unit displacements.



Releasing First Translation ( ):

1) Find the coefficients, using statics and mechanics of materials. From statics, one can

write three equations of equilibrium:

2) To find the values of k11, recall the principles learned from mechanics of materials:



Releasing Second Translation ( ):

1) Find the coefficients, using statics and mechanics of materials. From statics, one can

write three equations of equilibrium:

2) To find the values of k12 and k32, let’s examine the geometry of the deformed shape:



Releasing Third Translation ( ):

1) Similar to releasing the first translation, the coefficients for the third column of the

member stiffness matrix can be shown to be:

Releasing Fourth Translation ( ):

1) Similar from releasing the second translation, the coefficients for the fourth column of the

member stiffness matrix can be shown to be:

Assembling the Member Stiffness Matrix:

1) Subsittution of these values into the stiffness matrix, the __________________________

_____________________________________ is summarized as:

2) Note that the ith column of the member stiffness matrix, [k], consists of the end forces

required to create a unit displacement for , while all other translations are restrained.

3) The stiffness matrix is symmetric, that is .

4) The stiffness matrices for linear elastic structures are always symmetric.



Need for Coordinate Transformations:

1) Typically the members of a structure are oriented in _____________________________.

2) A need arises to transform stiffness relationships for each member to a common or

_______________________________________________.

3) ___________________________ must occur from a local coordinate system to a global

coordinate system.

4) This allows for a structure to be analyzed due to the _____________________ coordinate

system for all considered members.

Transformation Procedure (from Global to Local Coordinate Systems):

1) Consider again the planar truss as illustrated below and let’s focus on member ______.

2) Orientation of member m relative to the global coordinate system is given by _______.

a. ________ is measured ____________________ from the ___________________

_____________________________ of the global X axis to the local x axis.

Figure 8. Plane truss orientation in the local and global coordinate systems.



3) Recall the local stiffness relation:

4) Now, let’s construct relationships of the member end forces and displacements in the

global coordinate system.

a. Establish method whereby Q and u (from the ____________________________)

are mapped to F and v (to a ____________________________).

b. The equivalent system of forces and displacements in the global system can be

found by using geometry and an algebraic sum of the components.



Figure 9. Plane truss member m in local coordinates.

Figure 10. Plane truss member m in global coordinates.



5) At end b of the member m, the local forces ____ and ____ are equal to the algebraic sum

of the global forces in the direction of each local axis.

a. For the local x-axis, this can be expressed as:

b. For the local y-axis, this can be expressed as:

6) Using similar geometry and algebraic sums, at end e of member m, the relationship

between the local forces and the global forces is:

7) Rewriting this equation in matrix form:

8) Or symbolically as:



9) Where the transformation matrix, [T] is:

10) The cosines and sines can be determined using the nodal coordinates as:

11) The member end displacements are defined in the same manner in the global coordinate

systems. Therefore using the same transformation matrix, [T]. this relationship is:



Transformation Procedure (Local to Global Coordinate Systems):

1) In a similar manner from the transformation from global to local coordinate systems, the

procedure for the local to global coordinate systems is defined below.

2) At end b of the member m, the global forces ____ and ____ are equal to the algebraic

sum of the local forces in the direction of each global axis.

a. For the global x-axis, this can be expressed as:

b. For the global y-axis, this can be expressed as:

3) Using similar geometry and algebraic sums, at end e of member m, the relationship

between the global forces and the local forces is:

4) Rewriting this equation in matrix form:



5) Furthermore by comparison of this equation and the previous one expressing the reverse

relation, one key matrix property of [T] is _____________________________________:

6) The member end displacements are defined in the same manner as the corresponding

forces. Therefore the transformation of the member end displacements from the local to

global coordinates system is:

Member Stiffness in Global Coordinate Systems:

1) Let’s combine stiffness relations for the plane truss members in local coordinates with the

transformation matrix to a member stiffness matrix in aligned global coordinates.

2) Knowing the previously outlined relationships for the local stiffness and the force

transformation matrix:

3) Which can be simply written as:



4) Where [K] is termed the member stiffness matrix in global coordinates.

5) Similar to the member local stiffness matrix [k]:

a. [K] is ____________________________________________________________

_________________________________________________________________.

b. The physical meaning of a global stiffness coefficient, Kij: a stiffness coefficient,

_______, represents the _____________ at the location and direction of _______

required, along with other end forces, to cause a __________________________,

______, while all other displacements _______ are zero (or null).

c. Refer to the text for details on the derivation of the global stiffness matrix, [K] by

imposed displacements for each DOF in the global coordinate system.

Structure Stiffness Relations in Global Coordinate Systems:

1) The stiffness relationship for the _____________________________________________

is required to analyze a truss structure.

a. Use a combination of member stiffnesses (as appropriate).

2) Recall the relationship as:



3) General procedure:

a. Express the ________________________, [P], in terms of end forces in global

coordinates, [F]. Apply equilibrium for each node.

b. Express the __________________________________________,{d}, as related to

member end displacements in global coordinates,{v}. Apply compatibility

conditions to relate member end displacements to joint displacements.

c. Substitute ___________________________________________ into the member

force relation, ___________________, to express the global end forces in terms of

joint displacements. These ______________ relations are then substituted into the

joint equilibrium to establish the structure stiffness relationship (______________).

4) To make this procedure more apparent, let’s consider an example:

a. A truss is identified below in the figure. This truss has _________ degrees of

freedom, identified as _______ and _______.

b. Note the orientation of members 1 and 3 differ than that of member 2. This helps

illustrate the formulation of the structure stiffness matrix, [S].



Figure 11. Example truss subjected to arbitrary loading at the top node.

Figure 12. Discretization of example truss.



5) Applying the three general steps to obtain the structure stiffness method:

6) Equilibrium: apply equilibrium equations to the ________________.

Figure 13. Joint equilibrium at node ______.



7) Compatibility: inspect the boundary conditions, member ends, and joint displacements of

each member.

Figure 14. Member compatibility conditions for the example truss.



8) Member stiffness relationships: link the aforementioned relationships (_______________

and __________________________) and write the global stiffness equation:

9) In expanded matrix form, for member 1:

10) Where the forces at end 2 of the member 1 are:

11) In like fashion, the forces at the end 1 of member 2 are:

12) In like fashion, the forces at the end 2 of member 3 are:



13) Structure Stiffness Relations: through substitution of the aforementioned equations, it can

be shown that the joint loads, [P], and the joint displacements, {d}, of the plane truss are:



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

termed the ___________________________________________.

15) [S] is termed the ______________________________________, which is a square

matrix of order equal to the number of degrees of freedom.

a. [S] is ____________________________________________________________

_________________________________________________________________.

b. The physical meaning of a structure stiffness coefficient, Sij: the _____________

at the location and direction of _______ required, along with other end forces, to

cause a ___________________________ value at ______, while all other

displacements are ___________________.

c. [S] can be also be determined by imposing unit valued displacements for each

DOF and evaluation of the joint loads (see text from more details).

Structure Stiffness Matrix and Support Reactions using Code Numbers:

1) Formulation of [S] and [R] by using a _________________________________________

______________ to correctly position and assemble elements.

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

be _____________________________________________________________________.

3) To illustrate the procedure, let’s examine the previous truss example.



Figure 15. Discretization of example truss.

Figure 16. Member compatibility conditions for the example truss.



4) When establishing the code numbers, for a member stiffness, [K], the orientation of the

member will dictate the end joints.

a. Otherwise known as the location of node i (beginning) and node j (end).

5) Apply the compatibility equations to member 1, the code numbers correspond to 3, 4, 1,

and 2. Therefore this implies:

6) The member code numbers can also be utilized to construct the joint and reaction vectors.

Writing the code numbers for each member in regard to their respective force vectors:

7) Therefore one can see that the equilibrium at the joints provides the joint loads:



8) To illustrate the combination of the structure stiffness matrix, let’s partition each member

stiffness matrix using _______________________________:

9) After partitioning the matrices, an algebraic sum of the relevant elements in their aligned

position will produce the structure stiffness matrix.



10) The support reaction vector, [R], can be constructed in like fashion using code numbers.

a. This is done using equilibrium at the nodes.

b. When appropriate, global end forces of the members are algebraically summed to

represent the restraint.

c. For the example truss, this is illustrated as:



General Procedure for Analysis for Plane Trusses:

1) Discretize analytical model.

a. Draw and label line diagram.

b. Establish the _______________________________________________ systems.

c. Identity the _______________________________ and the restrained coordinates.

2) Construct the structure stiffness matrix, [S].

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

i. Determine local stiffness matrix, _______.

ii. Apply the transformation matrix, _______.

b. Identify the code numbers and assemble the elements of [S].

i. Alternatively can use direct stiffness approach or other methods.

ii. When using the direct stiffness approach, the entire structure stiffness

matrix can be determined, _______.



iii. When using the entire structure stiffness matrix, it is necessary to partition

the relationship to solve for DOF:

3) Formulate the joint load vector, _______.

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

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

displacements, and support reactions as needed.

6) Check equilibrium to verify the solution.



Plane Trusses: Examples

Example #1

Determine the joint displacements, axial forces in each member, and the reactions for the truss

structure shown below. Use the matrix stiffness method.

E = 70 GPa A = 2,000 mm2

Solution:

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

Identify the number of reactions. NR = ________.

E = _____________.

A = _____________.



Finding the stiffness matrix for member 1.

L =

cosθ =

sinθ =

Finding the stiffness matrix for member 2.

L =

cosθ =

sinθ =



Assembling the Structure Stiffness Matrix.

Assembling the Joint Load Vector.

Find the Joint Displacements.



Compute Member End Displacements and Forces.

Compute the Reactions.



Example #2

Determine the joint displacements, axial forces in each member, and the reactions for the truss structure shown below. Use the matrix stiffness method.

E = 200 GPa A1 = 4,000 mm2

A2 = 4,000 mm2 A3 = 6,000 mm2

Documents

Plane Trusses - University of Nebraska–Lincoln. Plane Trusses.pdfPlane Trusses Notes prepared by: R.L. Wood Page 1 of 40 Plane Trusses Lesson Objectives: 1) Identify the nomenclature