Analysis of Plane Frames


 Jeremy Malone
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1 Plane frames are twodimensional structures constructed with straight elements connected together by rigid and/or hinged connections. rames are subjected to loads and reactions that lie in the plane of the structure. Analysis of Plane rames Under the action of external loads, the elements of a plane frame are subjected to axial forces similar to truss members as well as bending moments and shears one would see in a beam. ence the analysis of plane frame members can be conveniently conducted by treating the frame as a composite structure with beam elements that can be subject to axial loads. These elements are usually rigidly connected or semirigidly connected depending on the amount of rotational restraint designed into the connection. D rame Moment connections throughout
2 Equivalent Joint oads The calculations of displacements in larger more extensive structures by the means of the matrix methods derived later requires that the structure be subject to loads applied only at the joints. Thus in general, loads are categorized into those applied at joints, and those that are not. oads that are not applied to joints must be replaced with statically equivalent loads. Consider the statically indeterminate frame with a distributed load between joints B and C:
3 The frame on the previous page is statically equivalent to the following two structure: The frame to the left is statically equivalent to the original frame in the sense that the reactions would be the same. owever the internal bending moment in the cross beam would not be the same. The frame would not be kinematically consistent with the original frame. owever, the reactions at the foundations and the joint actions would be equivalent. Keep in mind we are trying to find these unknown actions (forces and moments).
4 Example 8.1 Redundants are forces and moments Zero and nonzero displacements in the released structure due to external loads are shown The plane frame shown at the left has fixed supports at A and C. The frame is acted upon by the vertical load P as shown. In the analysis account for both flexural and axial deformations. The flexural rigidity EI is constant. The axial rigidity EA is also constant. Joint B is a rigid connection and we will endeavor to preserve equilibrium at Joint B and throughout the structure. The structure is statically indeterminate to the third degree. A released structure is obtained by cutting the frame at joint B and the released actions Q 1, Q and Q are redundants. ind the magnitude and direction of these redundants.
5 The displacements in the released structure caused by P and corresponding to Q 1, Q and Q are depicted in the previous figures. These displacements in the released structure caused by the external load are designated D Q1, D Q and D Q respectively. The displacement D Q1 consists of the sum of two translations which are found by analyzing the released structure as a set of two cantilever beams AB and. irst analyze the cantilever beam AB. The load P will cause a downward translation at B and a clockwise rotation at B. There is no axial displacement thus D Q 1 AB The displacements D Q and D Q in the released structure AB consist of a vertical displacement and a rotations, i.e., D Q AB 5P 48EI D Q AB P 8EI But these displacements in the released structure AB have corresponding displacements in the released structure.
6 owever, since there is no load on member in the released structure, there will be no displacement at end B and Even though the displacements at B in are zero, the total displacement of joint B would be the summation of the two D Q components from AB and. Thus the D Q matrix is D Q D D D Q1 Q Q 5P 48EI P 8EI 5 6 We need to assemble the flexibility matrix. Consider the released structure with Q Q 1 1 Q P 48EI
7 The displacements at end B of member BC are The displacements corresponding to unit values of Q 1, Q and Q are shown as flexibility coefficients 11, 1 and 1. If both axial and flexural deformations are considered, the displacements at end B of member AB are AE AB AB AB EI EI The flexibility coefficients take on the following values AE 11 EI 1 1 EI
8 The same analysis must be made with Q Q 1 1 Q or frame sections AB and AB : EI 1 AB AB AB EI AE : 1 Which leads to flexibility coefficients 1 EI AE EI
9 Once again from frame sections AB and (Q = 1, Q 1 = Q = ) AB : EI 1 AB AB AB EI : EI 1 EI leading to flexibility coefficients 1 EI EI EI EI
10 Assembly of the flexibility matrix leads to EA EI EI EI EA EI EI EI EI EI Now let P 1 K 1 ft 144 inches E I, in 4 ksi A 1 in
11 When these numerical values and D Q EA EI EA 1 144, 144,.48 in / kip in / kip The axial compliance (flexibility) of each component is quite small relative to the flexural compliance (flexibility). We will ignore the axial compliance of the beam and the column when assembling the flexibility matrix. The inverse of compliance is stiffness. This is equivalent to stating that the axial stiffness of the beam and column is so large relative to flexural stiffness of the beam and column that axial displacements are negligible.
12 Omitting axial deformations leads to the following flexibility matrix When the flexibility matrix is inverted we obtain
13 With 1 and D Q we can compute the unknown redundants utilizing the matrix equation i.e., 1 Q D Q D Q Q kips 4.69 kips kip inches Note that the displacements associated with the redundants in the original structural, represented by the matrix {D Q } are zero because joint B is a rigid connection. One can rationalize the rotation D Q is zero from this assumption.
14 Now examine the structure by omitting axial deformations. The flexibility matrix inding the inverse of this matrix (homework assignment) and substitution into 1 Q D Q D Q leads to Q.9167 kips kips 88.6 kip inches which is less then % different from the computations where axial deformations are included. This frequently happens in the analysis of typical frames and bolsters the assumption that D Q1 and D Q (axial deformations associated with axial force redundants) are zero. This allows the seasoned engineer to make judgments about considering only bending in frame analyses.
15 Example 8.
16 Example 8.
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