Structural Analysis I - SNU

1 Dept. of Civil and Environmental Eng., SNU Structural Analysis I Spring Semester, 2015 Hae Sung Lee Dept. of Civil and Environmental Engineering Seoul National University y yf z zf x xf yM y zM z xM x Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU This page is intentionally left blank. Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU Contents 1. Introduction 2. Reactions & Internal Forces by Free Body Diagrams 3. Principle of Virtual Work 4. Analysis of Statically Indeterminate Beams 5. Analysis of Statically Indeterminate Trusses 6. Analysis of Statically Indeterminate Frames 7. Influence Lines for Determinate Structures 8. Influence Lines for Indeterminate Structures Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU This page is intentionally left blank. Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng.

2 , SNU 1 Chapter 1 Introduction Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 2 Mechanics of Material - Structural Mechanics Problem Calculate the reaction force at each support and draw the moment and shear force diagram for the two-span beam shown in the figure. Solution Equilibrium Equation qLRRRF cbay20=++ = qLRRLRLRLqLMcbcba220220=+ = = 00220= = + + = cacabRRLRLRLqLLqLM qLRRLRLRLqLMbabac220220=+ = + + = Since there are three unknowns in two independent equations, we cannot determine a unique solution for the given structure, and thus we need one more equation to solve this problem. The main issue of this class is how to build additional equations to analyze statically indeterminate structures. EI EI q Ra Rb Rc L L q Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 3 Mechanics of Material Governing Equation Left span 11213141''''124dxcxbxaEIqxwqEIw++++= = Right Span 22223242''''224dxcxbxaEIqxwqEIw++++= = Boundary Conditions Left support 0)0()0( , 0)0(111= ==wEIMw Center support )()( , )()( , 0)()(212121 LwLwLwLwLwLw = = == Right support 0)0()0( , 0)0(222= ==wEIMw Since there are eight unknowns with eight conditions, we can solve this problem.

3 Determination of Integration Constant Left Support xcxaEIqxwbwdw1314111112402)0( , 0)0(++= == == Right Support xcxaEIqxwbwdw2324222222402)0( , 0)0(++= == == x x q w1 w2 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 4 Center Support == == +=+ =++=++==++=EIqLccEIqLaaLaEIqLLaEIqLcLaEI qLcLaEIqLLcLaEIqLLwLcLaEIqLLw48483626236 36024)(024)(321212212222312132324213141 )32(4833421xLLxxEIqww+ = 83 , 83211211qLqxwEIVxqLxqwEIM+ = =+ = = Moment Diagram Shear Diagram Reactions L83 + - + - L83 + - + Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 5 Mechanics of Material + Main idea Original Problem Case I (Removal of the center support) Case II (Application of the reaction force) Original Problem = Case I + Case II 0+ R=0 (compatibility condition) Calculation of 0 Governing Equation dcxbxaxEIqxwqEIw++++= =2340''''024 q q 0 R Rb Structural Analysis Lab.

4 Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 6 Boundary (support) Conditions Left Support 0)0()0( , 0)0(000= ==wEIMw Right Support 0)2()2( , 0)2(000= ==LwEILMLw Determination of Integration Constant Left Support 00)0( , 0 0)0(00= = = =bwdw Right Support += = =+=++ = =EILqcEILqaLaEILqLcLaEILqLwLw24)2(12)2(0 )2(62)2(0)2()2(24)2(0)2( 0)2(323400 ))2()2(2(243340 LxLxxEIqw+ = EILqLLLLLEIqLw384)2(5))2()2(2(24)(433400 =+ == Calculation of R Governing Equation dcxbxaxwEIwRR+++= =23''''0 Boundary (support) Conditions Left Support 0)0()0( , 0)0(= ==RRRwEIMw Mid-span 2)()( , 0)()(bRRRRLwEILVLwL = == = Determination of Integration Constant Left Support 00)0( , 0 0)0(= = = =bwdwRR Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 7 Mid-span )3(121231226032)( 0)( 2322xLxEIRwEILRcEIRaRaEIcaLRLwEILwbRbbbb RR = == ==+ = = EIRLEIRLLwbbRR48)2(122)(33 = == Final Solution Reaction at Supports qLRRRcba2=++ qLRRcb22=+ 0+ R=0 048)2(384)2(534= EIRLEILqb qLRb810= qLRRca83== Moment xqLxqxqLxLxqwEIwEIMMMRR83285)2(22200+ = + = =+= Shear 8385)(00qLqxqLLxqwEIwEIVVVRR+ = + = =+= Structural Analysis Lab.

5 Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 8 Structural Mechanics Original Problem Case I (Removal of the center support) Case II (Application of the reaction force) Original Problem = Case I + Case II 0+ R=0 Principle of Virtual Work EILqdxEIMMLR384)2(542000 == , EILRdxEIMMbLRRR48)2(320== Solution 0+ R=0 048)2(384)2(534=+ EIRLEILqb qLRb810= RbL/2 Rb q q qL2/2 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 9 Moment Shear + - + - + + - + - = 5qL2/8 Rb + = L83 + - + qL2/2 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 10 (Supports) (fixed support) Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 11 (hinge support) (roller support) Structural Analysis Lab. Prof. Hae Sung Lee, Dept.

6 Of Civil and Environmental Eng., SNU 12 2 (Main Structure) (Cross Beam) (Stringer ) Cross Bracing (Wind Bracing) (Support) Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 13 Truss Beam (Joint) Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 14 Frame Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 15 Force and Displacement Real 3-D Structures 3 force components and 3 moment components 3 displacement components and 3 rotational components Beam Idealization Vertical force and Moment on z-axis Vertical displacement and rotational angle z-axis Plane Truss Idealization Vertical and horizontal force Vertical and horizontal displacement x y z y yf z zf x xf yM y zM z xM x 22 ,wV 22 , M 11 ,wV 11 , M 33 ,wV 33 , M Structural Analysis Lab.

7 Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 16 Plane Frame Idealization Vertical, horizontal force and moment z-axis Vertical, horizontal displacement rotational angle z-axis x xf y yf x xf y yf zM z Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 17 (Stability of Structures) (Internal Stability) (External Stability) Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 18 This page is intentionally left blank. Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 19 Chapter 2 Reactions & Internal Forces by Free Body Diagrams Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 20 Free Body Diagram Structural Analysis Lab.

8 Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 21 It is impossible to draw too many free-body diagrams. Time spent in doing so is never wasted - C. H. Norris & J. B. Wilbur & S. Utku - Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 22 Reactions Beams 0 0= + = PRRFBAV 00= = LRPaMBA (Clockwise +) PLaRB= , PLbRA= PRRFBAV=+ = 0 0)(0= + = LRaLPMBA (Clockwise +) PLaRB)1(+= , PLaRA = L P a b RA RB L P a RA RBaL Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 23 Truss PHPHFAAH = =+ = 00, 00= + = PRRFBAV 0)3(0= + = aRPaPaMBA (Clockwise +) PRA31= , PRB32= Frame 002=+== += BAHBAVHHFqLRRF = =02 LHLRMBBRh = =0422 LqLLHLRMAALh BBHR2 = , 42qLHRAA+= 82242qLHHqLHqLHBABA= = + 1616qLHqLHBA == 883qLRqLRBA== P RA RB P HA L L HA HB RA RB q Structural Analysis Lab. Prof. Hae Sung Lee, Dept.

9 Of Civil and Environmental Eng., SNU 24 Internal Forces in Framed Structures Axial Force Shear Force Bending Moment Torsion + + + + Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 25 (Beam) Reactions q RA=qL/2 Rb= qL/2 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 26 Free Body Diagram for Shear and Moment qxqLqxRVVqxRFAxxAV = = = = 20 22022qxxqLMMxqxxRMxxAx = = = Shear Force and Moment Diagrams q RA=qL/2 RB= qL/2 RA RB RA x RB Mx Vx + qL/2 qL2/8 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 27 Deflected Shape Gerber Systems Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 28 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng.

10 , SNU 29 Internal Forces in a Gerber Beam - I Free Body Diagram PRLPLRMHHC3202430= = = PRPRRFCCHv3100= = + = PRLRLRMBHBA650450= = + = PRPRRFABAv610320 = = + = L/4 P RH RA RB RC P P/6 5P/6 P/3 P Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 30 Shear Force i) Lx 0 ii) LxL23 iii) LxL223 Bending Moment i) Lx 0 P/6 V= -P/6 P/6 5P/6 V= 2P/3 P/6 5P/6 P V= -P/3 + - - 2P/3 P/3 P/6 P/6 P/6 Mx= Px/6 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 31 ii) , iii) Deflected Shape P/6 5P/6 2P/3 Mx= P/6 5P/6 P P/3 P/3 Mx= + - PL/6 PL/6 Structural Analysis Lab. Prof. Hae Sung Lee, Dept. of Civil and Environmental Eng., SNU 32 Internal Forces in a Gerber Beam - II Free Body diagram Shear Moment q 2ql 2ql 2ql 2ql 22ql 22ql 2ql 2ql + L q L L 82ql 22ql 22ql + Structural Analysis Lab.