ZKC CTLR DESCRIBED CONFLICT EVENT AT FL320; BUSY TFC; FATIGUE; COMBINED POSITIONS; MULTIPLE WX DEVS AND STAFFING LISTED AS CONTRIBUTORY.

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THE PROX EVENT OCCURRED WHILE I WAS WORKING AN EXTRA (8 HR) SHIFT IN ADDITION TO MY NORMAL 5 DAY/40 HR WORK WK. THIS WAS THE SECOND WK OF 3 CONSECUTIVE 6 DAY WORK WKS. THERE WAS ADDITIONAL COMPLEXITY TO THE TFC DUE TO SIGNIFICANT WX/TSTMS IN THE AREA. A REQUEST HAD BEEN MADE FOR ADDITIONAL STAFFING ON THIS SHIFT FOR WX; BUT HAD BEEN DENIED. I WAS WORKING ALL OF OZARK SPECIALTY AREA (7 SECTORS -- INCLUDING LOW ALT; HIGH ALT; AND ULTRA-HIGH ALT STRATUMS) COMBINED AT 1 SCOPE. THERE WERE 3 SIGNIFICANT AREAS OF WX AFFECTING MY AIRSPACE AND TFC: 1) BTWN MCI AND ANX WITH MODERATE-EXTREME PPT; NO TOPS RPTS; BUT ACFT DEVS BTWN FL300-FL360. 2) JUST N OF SGF WITH MODERATE-EXTREME PPT; TOPS ON THE E RPTED AT FL400; TOPS ON THE W RPTED AT FL360; WITH MULTIPLE DEVS. 3) BTWN STL AND FAM 210 040 WITH MODERATE-EXTREME PPT; SOMEWHAT SCATTERED; NO TOPS RPTS; BUT MULTIPLE DEVS. AT THE TIME OF THE PROX EVENT; I WAS WORKING BTWN 15-17 ACFT WITH NUMEROUS DEVS AROUND THE AFOREMENTIONED WX. ACFT 2 CHKED ON AT FL320 REQUESTING DEVS AROUND THE WX NEAR SGF. ADVISED ACFT 2 TO STAND BY AS HE WAS STILL IN ZME AIRSPACE. ACFT 1 CHKED ON CLBING TO FL340. ONCE INSIDE MY AIRSPACE; ACFT 2 WAS CLRED TO DEVIATE UP TO 20 DEGS R FOR WX. ACFT 2 REQUESTED TO DEVIATE ON THE S SIDE OF THE WX. ACFT WAS THEN CLRED TO DEVIATE UP TO 20 DEGS L FOR WX. ACFT 1'S ALT WAS THEN AMENDED TO FL310 ONLY FOR POSSIBLE CONFLICTIONS WITH ACFT 2 AND EXPECT HIGHER SHORTLY. I PUT UP A J-RING ON ACFT 1. AFTER A FEW MINS; ACFT 2 ADVISED HE WAS THROUGH DEVIATING AND BACK ON COURSE. AFTER DEALING WITH A FEW OTHER DEV REQUESTS; I RE-EVALUATED A POSSIBLE CLB FOR ACFT 1. THE J-RING WAS STILL UP FOR ACFT 1 AND THE 2 ACFT APPEARED TO BE ABOUT 12 MI APART AND ON PARALLEL COURSES. ACFT 1 WAS REISSUED A CLB TO FL340. AT SGF; ACFT 1 BEGAN A TURN OF APPROX 50 DEGS TO LMN. NOTICING THE SIGNIFICANT TURN; I IMMEDIATELY ISSUED VECTORS TO EACH ACFT. ACFT 1 WAS ISSUED A TURN OF 20 DEGS L BUT FAILED TO RESPOND. ACFT 2 WAS THEN ISSUED TO EXPEDITE A TURN OF 20 DEGS R. ACFT 1 WAS REISSUED TO EXPEDITE A TURN 20 DEGS L FOR TFC. ACFT 2 WAS THEN ISSUED TO IMMEDIATELY TURN AN ADDITIONAL 30 DEGS R. PER ESAT; THE LOSS OF SEPARATION OCCURRED WITH 4.85 MI BEING THE CLOSEST DISTANCE OF ACFT 1 AND ACFT 2. CONTRIBUTING FACTORS: WX; ACFT DEVS; UNUSUAL RTES WITH SIGNIFICANT TURNS DUE TO WX; STAFFING; AND USE OF VECTORS LINES INSTEAD OF RTE LINES. AN ADDITIONAL PERSON ON THE SHIFT WAS WARRANTED DUE TO THE COMPLEXITY OF THE TFC AND WX. HUMAN FACTORS: I WAS TIRED. I WAS BUSY WITH WX PPT CALLS AND DEVS AND HAD NOT NOTICED THAT ACFT 1 WOULD MAKE A SIGNIFICANT TURN (APPROX 50 DEGS) AT SGF. I USED VECTORS LINES INSTEAD OF RTES LINES IN MY EVALUATION OF THE SITUATION. CORRECTIVE ACTIONS: I APPLIED IMMEDIATE VECTORS TO ENSURE SAFETY.