Plane Truss Stiffness Matrix

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Roy Blake
5 years ago
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1 ecture : Trusses & Grids tiffness Method Plane Truss tiffness Matri The distinguishing feature of a plane truss is that loads are applied in the plane of the structure whereas in a space truss they are not. We now wish to outline the procedure of formulating the joint stiffness matri [ J ] for a plane truss structure. The etension to a three dimensional space truss will be intuitively obvious.

2 ecture : Trusses & Grids tiffness Method Consider an arbitrary member, i. isolated from a generalized plane truss depicted below: y k i j Primary load carrying capabilities are in the plane of the truss hence the need for shear walls. The joints at the end of truss member i are denoted j and k. The plane truss lies in the -y plane. The joint translations are the unknown displacements and these displacements are epressed in terms of their and y components. We will rela the requirement that truss members are two force members. This allows for loads that are applied between joints to a truss member, and it allows consideration of the weight of the truss member.

3 ecture : Trusses & Grids tiffness Method The positive directions of the four displacement components (two translations at either end) of member i are depicted in the figure below It will be convenient to utilize the direction cosines associated with this arbitrary member. In terms of the joint coordinates the direction cosines are with C X cos k j C Y cos yk y y k j k j y j

ecture : Trusses & Grids tiffness Method The beam member stiffness matri developed in the previous section of notes can be easily adapted for use in the case of a plane truss.

4 ecture : Trusses & Grids tiffness Method The beam member stiffness matri developed in the previous section of notes can be easily adapted for use in the case of a plane truss. The member stiffness matri [ M ] for an arbitrary truss member with member aes X m and Y m oriented along the member and perpendicular to the member can be obtained by considering Case # and Case #7 from the previous section of notes. Using the numbering joint numbering system and the member aes depicted in the following figure then the member stiffness matri for a truss member is as follows

5 ecture : Trusses & Grids tiffness Method Note that [ J ] is based on aes oriented to the structure. Truss member stiffnesses may be obtained in one of two ways. Either the stiffnesses are directly computed using the figure to the left, or the second method consists of first obtaining the stiffness matri relative to the member oriented aes and then imposing a suitable matri transformation that transforms these elements to aes relative to the structure. We will focus on the direct method first to help develop an intuition of how the structure behaves. Unit displacement in both the and y directions are applied at each end of the member. If a unit displacement in the direction is applied to the j end of the member, the member shortens and an aial compression force is induced. The magnitude of the force is C 5

6 ecture : Trusses & Grids tiffness Method Restraint actions at the ends of the truss member in the and y directions are required. They are equal to the components of the aial force induced in the member, and are identified here as elements of the [ MD ] matri in order to distinguish them from elements of the [ M ] matri. The numbering of these elements are shown in the previous figure. Thus MD MD MD MD MD MD C C C y C C C y 6

7 ecture : Trusses & Grids tiffness Method In a similar fashion, a unit displacement in the y direction at the j end of the member yields MD MD MD MD MD MD C C C y y C C C y y 7

8 ecture : Trusses & Grids tiffness Method In a similar fashion, a unit displacement in the direction at the k end of the member yields MD MD MD MD MD MD C C C y C C C y 8

9 ecture : Trusses & Grids tiffness Method In a similar fashion, a unit displacement in the y direction at the k end of the member yields MD MD MD MD MD MD C C C y y C C C y y 9

10 ecture : Trusses & Grids tiffness Method We have just developed the four rows of the [ MD ] matri, i.e., 0

11 ecture : Trusses & Grids tiffness Method Eample. Consider the plane truss with four bars meeting at a common joint E. This truss only has two degrees of freedom from a kinematic standpoint. It is a convenience to identify the bars of the truss numerically. The bars have lengths,, and and aial rigidities,, and The loads consist of two concentrated forces P and P action at joint E. We will consider the bar weights identified here as w, w, w and w (force/length). The unknown displacements at joint E are identified as D and D. We seek to calculate member end actions A M, A M, A M and A M.

12 ecture : Trusses & Grids tiffness Method Because the weight of each truss member is included, the aial forces at either end of a truss member will be different at joints A, B, C and D then the aial force at joint E. The aial forces at joint E could be computed as well as the shear stresses at the end of each truss member, however they are omitted in this eample for simplicity. The loads P and P correspond to unknown displacements D and D, thus A D P P We net consider the restrained structure shown at the right. Here joint E is fied with a pin support that produce loads A D and A D associated with D and D.

13 ecture : Trusses & Grids tiffness Method Each truss member can be considered loaded as shown below. The points of support are indicated as A and E for the purpose of discussion and do not correspond to actual joints in labeled in the original truss. One could use the Greek alphabet, but the nomenclature should be transparent given the contet where it used.

14 ecture : Trusses & Grids tiffness Method ince the weights of the truss members produce no horizontal reactions, the actions A D must be zero and A D must be equal to half the weight of all the truss elements, i.e., A D w w 0 w w 0 W The quantity W is the total weight of the truss. For the purpose of calculating end actions for the vector A M, consider that from the previous figure A Mi w i i sin or A i M w w w w sin sin sin sin

15 ecture : Trusses & Grids tiffness Method The net step is formulating the stiffness matri by imposing unit displacement associated with D and D on the restrained structure as indicated below To obtain the stiffness values it is necessary to compute the forces in the truss elements when the unit displacements are applied to joint E. 5

16 ecture : Trusses & Grids tiffness Method P cos When the upper joint of the element moves to the right, the lower joint stays fied. P AE cos When the upper joint of the element moves up, again the lower joint stays fied. Both actions elongate the truss elements. The geometry of the elongation is determined by the translation of joint E. 6

17 ecture : Trusses & Grids tiffness Method When joint E is subjected to a unit translation to the right the truss element elongates an amount cos When joint E is subjected to a unit translation vertically the truss element elongates an amount sin The formulas given above are suitable for use in analyzing this plane truss. In a later lecture a more systematic approach to the development of member stiffnesses is developed that works for trusses and all types of structures. The stiffness is composed of contributions from various elements of the truss. Consider the contribution to from member, i.e., cos 7

18 ecture : Trusses & Grids tiffness Method Thus cos cos cos cos cos cos cos 0 cos The final epression results from the fact that truss element is horizontal and truss element is vertical. 8

19 ecture : Trusses & Grids tiffness Method imilarly the stiffness is composed of contributions from various elements of the truss. Consider the contribution to from member, i.e., Thus cos sin cos sin 0 cos sin cos sin cos sin cos sin cos sin cos sin 0 cos sin 9

20 ecture : Trusses & Grids tiffness Method By an analogous procedure and are cos sin cos sin sin sin The two epressions on this page as well as the two from the previous page constitute the stiffness matri []. The net step would be inverting this matri and performing the following matri computation to find the displacement D and D. D A D A D The vector {A D } and the matri {A D } were established earlier. 0

21 ecture : Trusses & Grids tiffness Method ince the vector {A M } was determined earlier as well, we need only identify the elements of the matri {A MD }. This matri contains the member end-actions due to unit displacements associated with the displacements D and D, but the end actions are computed using the restrained structure. Thus for i th member using a previous figure thus A i i MDi cos i AMDi sin i i i And we can now solve A MD cos cos cos cos sin sin sin sin A A A D M M MD

22 ecture : Trusses & Grids tiffness Method Eample.

23 ecture : Trusses & Grids tiffness Method tructural Grids (Need a theory section on - tiffness Method as it applies to grids Eamples of two dimensional grids are depected below: Grids behave more like frames. However, loads are typically applied perpendicular to the plane of the structure. Moments are allowed in the plane of the structure and perpendicular to the structure.

24 ecture : Trusses & Grids tiffness Method A space grid the structure is not in a plane, but the loads are in essence perpendicular to the structure.

25 ecture : Trusses & Grids tiffness Method Eample. The grid shown below consists of two members (AB and BC) that are rigidly joined at B. Each member is assumed to have fleural rigidity EI and torsional rigidity GJ. Kinematically, the only unknowns are the displacements at B. ince aial rigidities of the members is assumed to be quite large relative to EI and GJ, the displacements at B consist of one translation (D ) and two rotations (D and D ). Determine these unknown displacements. 5

26 ecture : Trusses & Grids tiffness Method When analyzing a grid by the stiffness method, an artificial restraint is provided at joint B, i.e., It is easier to see what the reactions are if we break the structure above into two substructures such that 6

27 ecture : Trusses & Grids tiffness Method From the last figure it is easy to see that A D P A D 0 A D P 8 A D 0 A 0 D AD 0 or A D P A D 0 A D P 8 and in a matri format P A D 0 8 7

28 ecture : Trusses & Grids tiffness Method The vector {A D } represents actions in the unrestrained structure associated with the unknown displacement D, D and D. ince there are no loads associated with these displacements {A D } is a null vector and in a matri format A D We have {A D } and {A D } the net step is the solution of the superposition epression D A D A D for the unknown displacements. To do that we need to formulate the stiffness matri and find its inverse. 8

29 ecture : Trusses & Grids tiffness Method The stiffness matri is found by analyzing the restrained structure for the effects of unit translations and rotations associated with the unknown displacements. In the following figure the grid structure is once again split into two substructures. From the figures above EI 0 6EI EI 6EI 0 EI 6EI 6EI 9

30 ecture : Trusses & Grids tiffness Method To obtain the second column of the stiffness matri utilize the following figure From the figures above GJ 0 0 6EI EI 0 6EI EI GJ 0 0

31 ecture : Trusses & Grids tiffness Method To obtain the third column of the stiffness matri utilize the following figure From the figures above 6EI 0 EI 0 0 GJ J 6EI 0 EI GJ

32 ecture : Trusses & Grids tiffness Method Define EI GJ EI C EIC C then and inverting this stiffness matri leads to 5 C C C where

33 ecture : Trusses & Grids tiffness Method olving leads to D A D A D D 96EI P

34 ecture : Trusses & Grids tiffness Method Eample.

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