Beams. Lesson Objectives:

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1 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. Beams Richard L Wood, 2017 Page 1 of 45

2 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 beam 1. 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. 2 nd edition. Cengage Learning. Beams Richard L Wood, 2017 Page 2 of 45

3 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. Beams Richard L Wood, 2017 Page 3 of 45

4 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. Beams Richard L Wood, 2017 Page 4 of 45

5 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. Beams Richard L Wood, 2017 Page 5 of 45

6 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. Beams Richard L Wood, 2017 Page 6 of 45

7 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: Beams Richard L Wood, 2017 Page 7 of 45

8 Figure 6. Derivation of stiffness coefficients by imposed unit displacements:. Figure 7. Derivation of stiffness coefficients by imposed unit displacements:. Beams Richard L Wood, 2017 Page 8 of 45

9 Figure 8. Derivation of stiffness coefficients by imposed unit displacements:. Figure 9. Derivation of stiffness coefficients by imposed unit displacements:. Beams Richard L Wood, 2017 Page 9 of 45

10 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: Beams Richard L Wood, 2017 Page 10 of 45

11 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:. Beams Richard L Wood, 2017 Page 11 of 45

12 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: Beams Richard L Wood, 2017 Page 12 of 45

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14 Second 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: Beams Richard L Wood, 2017 Page 14 of 45

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16 Third 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: Beams Richard L Wood, 2017 Page 16 of 45

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18 Fourth 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: Beams Richard L Wood, 2017 Page 18 of 45

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20 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. Beams Richard L Wood, 2017 Page 20 of 45

21 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: Beams Richard L Wood, 2017 Page 21 of 45

22 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. Beams Richard L Wood, 2017 Page 22 of 45

23 Figure 12. Example continuous beam structure subjected to arbitrary loading. Figure 13. Discretization of example beam. Beams Richard L Wood, 2017 Page 23 of 45

24 (a) (b) (c) (d) (e) Figure 14. Details on the member end forces (a, c, and e) and joints (b and d). Beams Richard L Wood, 2017 Page 24 of 45

25 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: Beams Richard L Wood, 2017 Page 25 of 45

26 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: Beams Richard L Wood, 2017 Page 26 of 45

27 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. Beams Richard L Wood, 2017 Page 27 of 45

28 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 (a) (b) Figure 15. Example continuous beam structure subject to. Beams Richard L Wood, 2017 Page 28 of 45

29 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. Beams Richard L Wood, 2017 Page 29 of 45

30 (a) (b) Figure 16. Example continuous beam structure subject to. Beams Richard L Wood, 2017 Page 30 of 45

31 3) By examining Figure 5, a unit displacement is induced at, while restraining 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. 4) By examining Figure 6, a unit displacement is induced at, while restraining 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. Beams Richard L Wood, 2017 Page 31 of 45

32 (a) (b) Figure 17. Example continuous beam structure subject to. 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). Beams Richard L Wood, 2017 Page 32 of 45

33 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: 6) By applying the compatibility equations to member 2, the code numbers correspond to:. Therefore this implies: Beams Richard L Wood, 2017 Page 33 of 45

34 7) By applying the compatibility equations to member 3, 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. Beams Richard L Wood, 2017 Page 34 of 45

35 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. Beams Richard L Wood, 2017 Page 35 of 45

36 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. Beams Richard L Wood, 2017 Page 36 of 45

37 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 Richard L Wood, 2017 Page 37 of 45

38 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 in 4 Solution: Identify the number of degrees of freedom. NDOF =. Identify the number of reactions. NR =. E =. I =. Beams Richard L Wood, 2017 Page 38 of 45

39 Finding the stiffness matrix for member 1. L = Finding the stiffness matrix for member 2. L = Beams Richard L Wood, 2017 Page 39 of 45

40 Assembling the Structure Stiffness Matrix. Assemble the Joint Load Vector. Find the Joint Displacements. Beams Richard L Wood, 2017 Page 40 of 45

41 Compute Member End Displacements and Forces. Compute the Reactions. Beams Richard L Wood, 2017 Page 41 of 45

42 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(10 6 ) mm 4 : Solution: Identify the number of degrees of freedom. NDOF =. Identify the number of reactions. NR =. E =. I =. Beams Richard L Wood, 2017 Page 42 of 45

43 Finding the stiffness matrix for member 1. L = Finding the stiffness matrix for member 2. L = Beams Richard L Wood, 2017 Page 43 of 45

44 Assembling the Structure Stiffness Matrix. Assembling the Joint Load Vector. Find the Joint Displacements. Beams Richard L Wood, 2017 Page 44 of 45

45 Compute Member End Displacements and Forces. Compute the Reactions. Beams Richard L Wood, 2017 Page 45 of 45

Slope Deflection Method

Slope Deflection Method Lesson Objectives: 1) Identify the formulation and sign conventions associated with the Slope Deflection method. 2) Derive the Slope Deflection Method equations using mechanics