THEORY OF STRUCTURE SSC-JE STAFF SELECTION COMMISSION CIVIL ENGINEERING STRUCTURAL ENGINEERING THEORY OF STRUCTURE

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1 Page 1 of 97 SSC-JE STAFF SELECTION COMMISSION CIVIL ENGINEERING STRUCTURAL ENGINEERING 28-B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

C O N T E N T Page 2 of 97 1. ETERMINACY AN INETERMINACY 3-19 2. ROLLING LOAS AN INFLUENCE LINES 20-36 3. ARCHES AN CABLES 37-51 4.

2 C O N T E N T Page 2 of ETERMINACY AN INETERMINACY ROLLING LOAS AN INFLUENCE LINES ARCHES AN CABLES ANALYSIS STATICALLY INETERMINATE STRUCTURES TRUSSES B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

3 Page 3 of 97 CHAPTER-1: ETERMINACY AN INETERMINACY Equations of Static Equilibrium (a) Plane Structure (2-structure): A structure is said to be 2- structure or plane structure when all the members or forces in the structure are in one plane onl. Some eamples of plane structures are: (i) Beams (ii) Plane trusses (iii) Plane frame (iv) Cables (v) Arches, etc. The equation of equilibrium for a planar structure are F 0 F 0 M 0 (b) Space Structure (3-structure): Some eamples are, (i) Space truss The equations of equilibrium are: F 0 F 0 F 0 A structure is said to be 3 structures in which members and forces are in 3. M 0 M 0 M 0 (ii) Space frame etc. Staticall eterminate Structures: A structure is said to be staticall determinate structure if the condition of equilibrium are sufficient to fullanale the structure. B.M. and S.F. at a section are independent of the material properties and cross-sectional dimensions of the components of the structure. No stresses are induced due to temperature changes and lack of fit. e.g.-simpl supported beam, cantilever beam etc. Staticall Indeterminate Structures OR Redundant Structures: A structure is said to be staticall indeterminate structure if the conditions of equilibrium aren t sufficient to full anale the structure. B.M. and S.F. at a section depends on the material properties and cross sectional dimensions of the components of the structure. Stresses are induced due to temperature changes and lack of fit. e.g. - Fied beam, continuous beam etc. 28-B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

4 Page 4 of 97 (A) STATIC INETERMINACY If there are n unknown like moment, shear, aial force etc. even after appling the laws of static equilibrium, the structure is said to be redundant to n degree. (i) Eternal Redundanc: If the eternal support reactions cannot be determinedd b using the equations of static equilibrium, the structure is termed as eternall redundant. It is equal to number of eternal reaction components in addition to number of equilibrium conditions. (ii) Internal redundanc: If the internal member forces provided to safel resist the eternal forces cannot be determined b using the equation of static equilibrium, the structure is teamed as internall redundant. It refers to geometric stabilit of the structure. Total Indeterminac: S Si Se egree of eternal static indeterminac or S Se egree of internal static indeterminac = No. of unknown forces Equations of static equilibrium available = Total no. of support reactions No. of equations of static equilibrium available = R 3 For 2 structure = R 6 For 3 structure Support Reactions: (a) Plane Structure : Support reactions are R, R and M (3 no.) Support reactions are R and R (2 no.) Support reaction is R (1 no.) 28-B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

5 Page 5 of 97 (b) Space Structure: Support reactions are R, R, R, M, M, M (6 no.) Support reactions are R, R, R, (3 no.) Support reaction is R (1 no.) Methods to determine static indeterminac of a structure. Case I: Pin jointed plane trusses: Trusses are pin-joined frames which carr onl aial forces. Static indeterminac (SI): SI= (m + r)-2j Where, m = Number of members r = Number of reactions j = Number of joints Total No. of unknowns = m + r Number of equations available at joint = 2j If SI = 0, Structure is staticall determinate SI> 0, Structure is staticall indeterminate SI< 0, Structure is kinematicall unstable For the figure given above, the static indeterminac (SI) is calculated as S.I. = m r 2 j e m = 7 r 3 j = 5 S.I. = = 0 Eternall redundant or indeterminate: for plane truss. For a planar structure, there are three equations of equilibrium H 0, V 0 & M 0 Hence, if Number of reactions = r then, If r> 3, Structure is eternall indeterminate r = 3, Structure is eternall determinate r< 3, Structure is eternall Kinematicall unstable (egree of eternal indeterminac) = r 3 Se Internall indeterminate: If m (2 j 3), Structure is internall indeterminate m (2 j 3), Structure is internall determinate m (2 j 3), Structure is internall kinematicall unstable. e 28-B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

6 Si (egree of internal indeterminac) = m (2j 3) Page 6 of 97 So, (Total degree of static indeterminac) = S Se Si = r 3 + m (2j 3) = m + r 2j If a rigid frame has hbrid joints such as presence of internal hinge, link, roller etc. than some of the internal reactions will be released hence si (egree of internal indeterminac) will be released. If r r is total number of released reactions then value of r r is calculated as rr m ' 1 { for 2 structure} r r = 3(m 1) {for 3 structure} m = number of members meeting at internal hinge. Tpes of Truss (i) Simple truss: When two bar and one joint are progressivel added to form a truss, the truss is called simple truss. (ii) Compound truss: These are the trusses formed b connecting two simple truss b a set of joints and bars. (iii) Comple truss: There is no joint in the truss where onl two bars meet. if hbrid joints are present then, = m + r 2j - r r = number of reactions released Case II: Rigid jointed plane frame: Unlike a pin jointed frame, in rigid jointed frame, a truss member resist three stress resultant (Aial, shear force and bending moment) Hence, [Total number of internal stresses = 3m] Total number of unknowns = 3m + r Also at ever joint 3 equations of equilibrium are available H 0, V 0, M 0 Total no. of equations available = 3j Staticall indeterminac (SI) SI (3) m 3 r j Note: This equation cannot be used as a generalied formula for all tpes of frame. In such cases SI = Total number of unknown Total number of equations available 28-B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

7 e.g. Page 7 of 97 Hence, If SI>0; Structure is staticall indeterminate SI = 0; Structure is staticall determinate SI<0; Structure is staticall kinematicall unstable Eternall indeterminate: If, r>3;structure is eternall indeterminate r = 0; Structure is eternall determinate r< 3 ; Structure is eternall Kinematicall unstable. se = r 3 Internall indeterminate: If, 3m> (3j 3); Structure is internall indeterminate 3m = (3j 3); Structure is internall determinate 3m< (3j 3); Structure is internall kinematicall unstable si = 3m (3j 3) If hbrid joints are present then, s 3m r 3 j r Where, r = Released reactions Ring Concept Let us take a general plane frame member. It ma be assumed as a ring subjected to loads and because of loads, it deforms. Therefore, internal forces are developed in the ring. Now, these internal forces can be found out b making a cut. A cut releases three internal forces, shear (V), aial force (P) and bending moment (M) at a section Hence, total unknown member forces = 3 Appling above concept for closed frames, static indeterminac can be calculated as, Si (Internal Indeterminac) = 3C r (For 2 structure) Si (Internal Indeterminac) = 6C r (For 3 structure) Se (Eternal Indeterminac) = R 3 (For 2 frames) Se (Eternal Indeterminac) = R 6 (For 3 frames) Where, C = Number of loops (rings) (M 1) (For 2) r r j 28-B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

8 3(M 1) M j j = Number of member connecting with j number of joints j = Number of hbrid joint. Case III: Pin Jointed Space Truss : In a 3-dimensional pin-jointed truss, all the members carr aial force onl and hence the number of total unknown internal forces = m and let total of reactions be r. Hence, total number of unknowns = m + r Number of equations available at joint 3 j F 0, F 0, F 0, Moment equations are automaticall satisfie ed Staticall indeterminac : SI m r 3 j Hence, [SI> 0, SI = 0, SI< 0 represents staticall indeterminate, determinate and Kinematicall unstable structure respectivel] If hbrid joints are present then S = m + r 3j - r r = number of reactions released Eternal Indeterminac: Number of equation of equilibrium is F 0 M 0 F 0 M 0 F 0 M 0 (For 3) si as given b: Hence, If, r>6; Structure is eternall ndeterminate r = 6; Structure is eternall determinate r< 6 ; Structure is eternall Kinematicall unstable Internal Indeterminac: If, m> (3j- 6); Structure is internall indeterminate m = (3j- 6); Structure is internall determinate m< (3j- 6); Structure is internall kinematicall unstable To bu Complete Course Materials Page 8 of 97 Contact: , Website : Classroom Coaching Program Postal Coaching Program Online Test Series 28-B/7, JiaSarai, Near IIT, HauKhas, New elhi Ph

Module 1 : Introduction : Review of Basic Concepts in Mechnics Lecture 4 : Static Indeterminacy of Structures

Module 1 : Introduction : Review of Basic Concepts in Mechnics Lecture 4 : Static Indeterminacy of Structures Objectives In this course you will learn the following Review of the concepts of determinate