RESEARCHES IN GRAPHICAL STATICS
CONTENTS 1 INTRODUCTION 1 CONCURRENT CO PLANAR FORCES 1 NON CONCURRENT CO PLANAR FORCES 6 CHAPTER 1 EXERCISES 15 2 GENERAL GUIDELINES FOR CREATING EF CIENT AND ELEGANT PROGRAMS 23 3 NON CONCURRENT FORCES AND THE FUNICULAR 31 CHAPTER 3 EXERCISES 48 4 THE FUNICULAR AND MOMENTS 57 CHAPTER 4 EXERCISES 78 5 TRUSS ANALYSIS AND DESIGN 95 INTERNAL MEMBER FORCES IN TRUSSES 100 METHOD OF SECTIONS 115 CHAPTER 5 EXERCISES 122 6 FRAMES 135 CHAPTER 6 EXERCISES 164 7 STABILITY IN 3D SPACE 179 DOFS OF A LINE IN 3D SPACE 180 STABILIZING A LINE IN 3D SPACE 181 THE FOUR NODE MODEL 182 THE NINE NODE MODEL 188 CHAPTER 7 EXERCISES 203 8 DE ECTIONS OF BEAMS AND INDETERMINATE BEAM ANALYSIS 213 INDETERMINATE BEAMS 221 CHAPTER 8 EXERCISES 237 9 INDETERMINATE TRUSS ANALYSIS 249 CHAPTER 9 EXERCISES 273 10 FORCES IN SPACE 291 CHAPTER 10 EXERCISES 313 INTRODUCTION 1 THIS TEXTBOOK WAS WRITTEN FOR STUDENTS AND PRACTITIONERS AND PROFESSORS WHO ARE INTERESTED IN LEARNING HOW TO USE GRAPHIC STATICS AFTER HAVING ALREADY STUDIED STATICS AND ELEMENTARY MECHANICS OF BEAMS THE HISTORY OF GRAPHIC STATICS IS A LONG AND RICH STORY WITH A BAF ING MYSTERY THROWN IN DURING THE MID TWENTIETH CENTURY MANY PAPERS HAVE RECENTLY BEEN WRITTEN DURING THE CURRENT RENAISSANCE OF GRAPHIC STATICS RESEARCH THAT IS OCCURRING TODAY THESE PAPERS NEATLY SUMMARIZE THE TRADITIONAL ARC OF THE STORY OF GRAPHIC STATICS STARTING WITH VARIGNON MOVING THROUGH CREMONA CULLMAN AND OTHERS GRAPHIC STATICS WAS PART OF EVERY STRUCTURAL ENGINEER S CURRICULUM UP TILL THE 1930S OR SO AND THOSE ENGINEERS BECAME COMFORTABLE WITH VERY ACCURATE DRAWING SOLUTIONS AND THEY WERE ABLE TO GRASP GEOMETRIC PRINCIPLES READILY YET AS DETAILED ALGEBRAIC SOLUTIONS BECAME MORE AND MORE WIDESPREAD IN THE MID TWENTIETH CENTURY AND CERTAINLY AS THE COMPUTER MADE INROADS INTO THE STRUCTURAL ENGINEER S TOOLKIT GRAPHIC STATICS RAPIDLY DISAPPEARED FROM ALMOST ALL STRUCTURAL ENGINEERING CURRICULA AROUND THE WORLD TODAY MOST STRUCTURAL ENGINEERS HAVE NO RST HAND KNOWLEDGE OF GRAPHIC STATICS EVEN IF THEY HAVE HEARD OF THE TECHNIQUE BECAUSE OF THE RECENT SPATE OF SCHOLARLY ARTICLES ON THE TOPIC THIS MYSTERY OF THE SUDDEN DEMISE OF GRAPHIC STATICS EDUCATION CAN BE PARTIALLY EXPLAINED BY THE RISE OF INTRICATE ALGEBRAIC METHODS AND THE RELIANCE ON COMPUTERS YET THAT DOES NOT FULLY EXPLAIN HOW SOMETHING SO BEAUTIFUL AND PROFOUND DISAPPEARED SO QUICKLY THE SCIENCE OF GRAPHIC STATICS IS AN EXACT METHOD FOR SOLVING TWO DIMENSIONAL AND THREE DIMEN SIONAL EQUILIBRIUM PROBLEMS THE POWER OF GRAPHIC STATICS MANIFESTS ITSELF WHEN SUCH ANALYSIS METHODS ARE THEN USED AS DESIGN METHODS BY MEANS OF MANIPULATING THE GEOMETRY OF THE PROBLEM SUCH ITERATIONS OF GEOMETRY ARE EXTREMELY RAPID IF THE DRAWINGS ARE DONE ON THE COMPUTER IN AN ENVIRONMENT THAT ALLOWS FOR MANIPULATION OF PARAMETERS IN CHAP 2 GENERAL GUIDELINES SUCH A DRAWING ENVIRONMENT IS INTRODUCED WITH SUGGESTIONS OF HOW TO ORDER AND FORMALIZE DRAWINGS CONCURRENT CO PLANAR FORCES THE STUDY OF GRAPHIC STATICS IN THIS TEXTBOOK BEGINS WITH CO PLANAR FORCES I E FORCES THAT CAN BE MODELED IN TWO DIMENSIONS A FORCE IS A VECTOR DISTINGUISHED BY THREE CHARACTERISTICS MAGNITUDE DIRECTION OF ACTION POINT OF APPLICATION THE STARTING POINT OF A VECTOR IS CALLED ITS ORIGIN AND THE END POINT WITH THE ARROW IS KNOWN AS THE TERMINUS IN FIG 1 1 TWO CO PLANAR FORCES ARE ACTING COMPRES SIVELY ON A POINT I E EACH VECTOR S TERMINUS TOUCHES THE SAME POINT THE FORCE THAT MAKES A KNOWN ANGLE WITH THE HORIZONTAL HAS A KNOWN MAGNITUDE OF F A SECOND FORCE IS TWICE THE MAGNITUDE OF 1 THE RST FORCE AND IT MAKES A KNOWN ANGLE TO THE HORIZONTAL WHAT FORCE R WOULD EQUILIBRATE THE POINT BEING LOADED BY F AND 2F THIS UNKNOWN FORCE IS SHOWN AS A DASHED VECTOR IN FIG 1 1 BUT OF COURSE ITS MAGNITUDE AND DIRECTION IS ONLY HINTED AT IN FIG 1 1 AS THEY ARE BOTH STILL UNKNOWN ALGEBRAICALLY OR GRAPHICALLY THIS PROBLEM HAS TWO UNKNOWNS THE MAGNITUDE AND DIRECTION OF THE EQUILIBRATING FORCE IT IS CLEAR THAT THE TERMINUS OF THE EQUILIBRATING FORCE MUST BE APPLIED TO THE POINT BEING LOADED BY F AND 2F THE EQUILIBRATING FORCE R MUST HAVE A MAGNITUDE EQUAL TO THE RESULTANT OF THE TWO APPLIED FORCES AND IT MUST ALIGN WITH THIS RESULTANT WITH AN OPPOSITE DIRECTION IT IS GOOD PRACTICE TO CONSIDER THE ORIGINAL PROBLEM AS THE FORM DIAGRAM I E A DEPICTION OF THE GEOMETRY OF THE ORIGINAL PROBLEM HERE THE FORM IS RATHER SIMPLE ONLY A POINT BEING LOADED BUT THE DIRECTIONS OF THE TWO LOADS IS ESSENTIAL INFORMATION THAT IS CAPTURED ON THE FORM DIAGRAM FORCE DIAGRAMS CORRESPONDING TO AN EQUILIBRIUM SOLUTION OF THE FORM DIAGRAM ARE SHOWN IN FIG 1 2 FIG 1 1 FORM DIAGRAM WITH TWO UNKNOWNS FIG 1 2 FORCE DIAGRAMS SHOW EQUIVALENT SOLUTIONS RESEARCHES IN GRAPHICAL STATICS 2 TWO SOLUTIONS ARE PROVIDED BECAUSE THE ORDER OF OPERATIONS DOES NOT MATTER THE TERMINUS OF F CAN BE APPLIED TO THE POINT AND THEN THE TERMINUS OF 2F CAN BE APPLIED TO THE ORIGIN OF F OR THE OPPOSITE ORDER CAN OCCUR THE TERMINUS OF 2F CAN BE APPLIED TO THE POINT AND THE TERMINUS OF F CAN BE APPLIED TO THE ORIGIN OF 2F EITHER WAY THE RESULTANT R IS FOUND DRAWING THE FORCE DIAGRAM TO SOME ARBITRARY SCALE IS A CRITICAL IDEA THE SCALE DOES NOT MATTER BUT THE FORCE VECTORS MUST BE DRAWN TO A KNOWN COMFORTABLE SCALE THE MAGNITUDE OF THE RESULTANT IS THE LENGTH OF THE DASHED LINE ON THE RIGHT SIDE OF FIG 1 2 THE FORCE DIAGRAM MULTIPLIED BY THIS SCALE FACTOR IN SUMMARY THE KEY IDEAS OF FIG 1 2 ARE THE FORM DIAGRAM ON ONE SIDE OF THE PAGE AND A FORCE DIAGRAM ON THE SAME PAGE BOTH DIAGRAMS ARE DRAWN TO INDEPENDENT SCALES BUT THIS INITIAL EXAMPLE HAD A SIMPLE FORM DIAGRAM SO ITS SCALE WAS NOT REALLY AN ISSUE FIGURE 1 3 DEMONSTRATES SOME OF THE POWER OF CREATING THESE GEOMETRIC SOLUTIONS IN A PROGRAMMA BLE DRAWING ENVIRONMENT CHANGING THE MAGNITUDE OF THE FORCE F AND CHANGING THE ANGLES AND PRESENTS NO DIF CULTY IN AN IMMEDIATE RE SOLVING FOR THE EQUILIBRATING FORCE R ANOTHER ELEMENTARY EQUILIBRIUM OF A POINT EXAMPLE SHOWS A FEW MORE GRAPHIC STATICS IDEAS IN THIS PROBLEM SHOWN IN FIG 1 4 ONE FORCE IS COMPRESSING A POINT ON THE GROUND THE QUESTION IS TO DETERMINE WHAT ARE THE X AND Y COMPONENTS OF THE EQUILIBRATING REACTION ONCE AGAIN CONSIDER THE ORIGINAL PROBLEM DRAWING TO BE THE FORM DIAGRAM AND CREATE A FORCE DIAGRAM TO SOME SUITABLE SCALE TO SOLVE THE PROBLEM IN THE RIGHT SIDE OF FIG 1 5 THE ORIGINAL FORCE VECTOR IS DRAWN TO A CONVENIENT SCALE AND A VERTICAL LINE IS PASSED THROUGH ITS TERMINUS AND A HORIZONTAL LINE IS PASSED THROUGH ITS ORIGIN FIG 1 3 IMMEDIATE UPDATE OF FORCE DIAGRAM WHEN INPUT IS VARIED FIG 1 4 UNKNOWN MAGNITUDES OF X REACTION AND OF Y REACTION RESEARCHES IN GRAPHICAL STATICS 3