Analyss of Contnuous Beams n General Contnuous beams consdered here are prsmatc, rgdly connected to each beam segment and supported at varous ponts along the beam. onts are selected at ponts of support, at any free end, and changes n cross secton occur at support ponts (.e., the beam s prsmatc. A contnuous beam havng m members and m+ jonts s depcted n fgure (a to the left. Support restrants of two types may exst ant any jont n a contnuous beam. These are restrants aganst rotaton and/or restrants aganst translatons. We wll only consder flexural deformatons. Torson and axal dsplacements are not consdered. d Thus only two dsplacements can occur at each jont.
Gven the numberng system n fgure (b the translaton at a partcular jont s numbered pror to a rotaton and t follows that the number of translatons s equal to the number of jonts mnus one, whle the rotaton s twce the jont number. Thus at jont j the translatons and rotaton are number 2j- and 2j respectvely. It s evdent that the total number of possble jont dsplacements s twce the jonts (or 2n j. If the total number of support restrants aganst translaton and rotatons s denoted n r, then the actual degrees of freedom are n 2n j 2m + n 2 r n r Here n s the number of degrees of freedom.
To relate the end dsplacements of a partcular member to the dsplacements of a jont, consder a typcal member n fgure (c below. The member end dsplacements are numbered j, j2, k and k3 and correspond to end dsplacements, 2, 3 and 4 n fgure (b. The new notaton helps facltate computer programmng. The four end dsplacements correspond to the four jont dsplacements as follows: j 2 j k 2 k j2 2 j k2 2k Snce j and k are equal numercally to and d(+, then: j j2 2 2 k k2 2 2 + + 2 Ths ndexng system s necessary to construct the jont stffness matrx [S j ]
The analyss of contnuous beams conssts of establshng the stffness matrx and the load matrx. The most mportant matrx generated s the overall jont stffness matrx [S j ]. The jont stffness matrx conssts of contrbutons from the beam stffness matrx [S m ]. It s convenent to assess the contrbutons for one typcal member and repeat the process for members through m. So the next step nvolves expressng the stffness coeffcents shown n the fgure to the left n terms of the varous member stffnesses the contrbute to the jont stffnesses.
Ths next step requres that the member stffnesses be obtaned from the matrx below: For example the contrbuton to the jont stffness (S j j, j from member - s the stffness S m33 for that member. Smlarly, the contrbuton to (S j j,j from member s the stffneess S m from member
In general the contrbuton of one member to a partcular jont stffness wll be denoted by appendng the member subscrpt to the member stffness tself. From ths dscusson one can see that the jont stffness matrx coeffcents are generated by the followng expressons: ( j j M 33 + S, M + j 2, j M 43 k, j M 3 M 4 k 2, j M 2 whch represent the transfer of elements of the frst column of the member stffness matrx [S m ]to the approprate p locaton n the jont stffness matrx [S j ]
Expressons analogous to the prevous expressons are easly obtaned for a unt rotaton about the z axs at jont j: ( j j M 34 + S, 2 M 2 + j 2, j 2 M 44 k, j 2 M 32 M 42 k 2, j 2 M 22 Expressons analogous to a unt y dsplacement at jont k are: 3 j, k 2 M j 2, k M 23 + k, k M 33 M + M 43 + M 2 + k 2, k
Fnally the expressons for a unt z rotaton at jont k are: 4 j, k 2 M j 2, k 2 M 24 (, M 34 + S k k M 2 + M 44 + M 22 + k 2, k 2 The last 4 sets of equatons show that the sxteen elements of the 4x4 member stffness matrx [S M ] for member I contrbute to the sxteen of the stffness matrx [S ] coeffcents n a very regular pattern. Ths pattern can be observed n the fgure on the next overhead.
For ths structure the number of jonts s seven, the number of possble jont dsplacements s fourteen, and the jont stffness matrx [S ] s dmensonally 4x4. The ndexng scheme s shown down the left hand edge and across the top. The contrbutons of ndvdual members are ndcated n the hatched block., each of whch s dmensonally 4x4. The blocks are numbered n the upper rght corner to ndcated the member assocated wth the block. The overlappng blocks are dmensonally 2x2 and denote elements that receve contrbutons from adjacent members.
Suppose that the actual beam has smple supports at all the jonts as ndcated n the fgure below. The rearranged and parttoned jont stffness matrx s shown at the lower rght. To obtan ths rearranged matrx, rows and columns of the orgnal matrx have been swtched n proper sequence n order t place the stffnesses pertanng to the actual degrees of freedom n the frst seven rows and columns. As an ad n the rearrangng process, the new row and column desgnatons are lsted n the prevous fgure for the matrx along the rght hand sde and across the bottom. The rearrangng process s consstent wth the numberng system n the fgure above.
In summary, the procedure followed n generatng the jont stffness matrx [S ] conssts of takng the members n sequence and evaluatng ther contrbutons one at a tme. Then the stffness matrx [S M ] s generated, and the elements of ths matrx are transferred to the [S ] as ndcated n the prevous overheads. After all members have been processed n ths manner, the [S ] matrx s complete. Ths matrx can be rearranged and parttoned n order to solate the [S] matrx. The nverse of ths matrx s then determned and the unknown dsplacements are computed.