WILLIOT EQUATIONS FOR STATICALLY INDETERMINATE STRUCTURES (IN COMBINATION WITH MOMENT EQUATIONS IN TERMS OF ANGULAR DISPLACEMENTS)

THE USEFULNESS OF THE WILLIOT DIAGRAM OF DEFLECTION EQUATIONS FOR STRUCTURAL ANALYSIS IS ILLUSTRATED. WHEN COMBINED WITH MAXWELL'S THEORUM OF RECIPROCAL DISPLACEMENTS, IT IS THE QUICKEST SOLUTION FOR STATICALLY INDETERMINATE REACTIONS IN CONTINUOUS SPANS, TWO-HINGED ARCHES, AND SIMILAR STRUCTURES. IT ALSO SERVES IN DETERMINING THE ANGULAR DISPLACEMENTS OF MEMBERS IN SECONDARY STRESS ANALYSIS. A PROBLEM WAS RECENTLY ENCOUNTERED IN STRESS DISTRIBUTION THAT WOULD NOT YIELD TO AN EXACT OR EVEN TO A SATISFACTORY SOLUTION BECAUSE OF THE LACK OF SUFFICIENT EQUATIONS, UNTIL THEY WERE DISCOVERED IN THE GEOMETRIC PROPERTIES OF THE WILLIOT DIAGRAM. THE DEVELOPMENT IS DESCRIBED OF THE WILLIOT STRAIN EQUATIONS AND IT IS SHOWN HOW THEY WERE COMBINED WITH MOMENT EQUATIONS EXPRESSED IN TERMS OF ANGULAR DISPLACEMENTS TO MEET THE SITUATION. THE EXPEDIENT IS DESCRIBED OF EXPRESSING MOMENTS IN TERMS OF ANGULAR DISPLACEMENTS AS THE VEHICLES BY WHICH SOLUTIONS CAN BE OBTAINED FOR SOME TYPES OF TOWERS FOR SUSPENSION BRIDGES FOR STATICALLY INDETERMINATE STRESSES. IF THE STRUCTURE IS STATICALLY INDETERMINATE, IN ORDER TO COMPUTE THE APPROXIMATE STRESSES REQUIRED FOR DESIGNING, IT IS NECESSARY TO FORMULATE A TENTATIVE DESIGN FOR MAKING CERTAIN ASSUMPTIONS AS TO THE PROBABLE BEHAVIOR OF THE STRUCTURE. THE ASSUMPTIONS MUST CLOSELY APPROXIMATE ACTUAL CONDITIONS. THE PATTERN OF WIND BRACING MAKES A TOWER FOR A SUSPENSION BRIDGE STRUCTURE MANY TIMES STATICALLY INDETERMINATE. THE DESIGN BEGINS AND PROCEEDS IN A SERIES OF FOUR APPROXIMATIONS FOLLOWED FINALLY BY A REVIEW. EACH APPROXIMATION INCLUDES A COMPUTATION OF STRESSES BASED ON CERTAIN ASSUMPTIONS, FOLLOWED BY A TENTATIVE DESIGN. AS ONE APPROXIMATION FOLLOWS ANOTHER THE ASSUMPTIONS WERE EITHER CORRECTED OR ELIMINATED SO THAT EACH TENTATIVE DESIGN APPROACHED MORE CLOSELY THE FINAL DESIGN. THIS PROCESS WAS CONTINUED UNTIL ALL ASSUMPTIONS EXCEPT ONE WERE ELIMINATED. SINCE THE EXTERNAL FORCES ACTING ON EACH POST WERE THE SAME, THEIR ELASTIC CURVES WERE ASSUMED TO BE IDENTICAL. THIS ASSUMPTION POSTULATED EQUAL BENDING MOMENTS AND SHEARS, EQUAL AND OPPOSITE AXIAL WIND STRESSES IN THE TWO POSTS, AND EQUAL VALUES OF ANGLES FOR CORRESPONDING JOINTS AND MEMBERS.

• Supplemental Notes:
  • Vol 100, PaPER NO 1904, PP 580-608, 16 FIG, 2 TAB
• Authors:
  • Ellis, C A
• Publication Date: 1935

Media Info

• Serial:
  • American Society of Civil Engineers Transactions
  • Publisher: American Society of Civil Engineers

Subject/Index Terms

• TRT Terms: Angularity; Approximation (Mathematics); Bending moments; Bracing; Bridge towers; Calculation; Deflection; Deformation curve; Dislocation (Geology); Equations; Mathematical analysis; Shear strength; Statically indeterminate frames; Structural analysis; Suspension bridges; Wind
• Uncontrolled Terms: Angles; Computation
• Old TRIS Terms: Statically indeterminate structure; Wind bracing
• Subject Areas: Bridges and other structures; Highways;

Filing Info

• Accession Number: 00208519
• Record Type: Publication
• Files: TRIS
• Created Date: Oct 14 1994 12:00AM