UNIT I. S.NO 2 MARKS PAGE NO 1 Why it is necessary to compute deflections in structures? 4 2 What is meant by cambering technique, in structures?

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1 1 UNIT I UNIT I INDETERMINATE FRAMES 9 Degree of static and kinematic indeterminacies for plane frames - analysis of indeterminate pin-jointed frames - rigid frames (Degree of statical indeterminacy up to two) - Energy and consistent deformation methods. S.NO MARKS PAGE NO 1 Why it is necessary to compute deflections in structures? 4 What is meant by cambering technique, in structures? 4 3 Name any four methods used for the computation of deflections in structures. 4 4 State the difference between strain energy method and unit load method in the determination of deflection of structures. 4 5 What are the assumptions made in the unit load method? 4 6 Give the equation that is used for the determination of deflection at a given point i in beams and frames. 4 7 The horizontal displacement of the end D of the portal frame is required. Determine the relevant equations due to the unit load at 5 appropriate point. 8 Due to load at B in the truss in fig. the forces in the members are as under. Determine the horizontal displacement of B by unit load method. 5 9 Determine the rotation of the curved beam in fig. due to a moment Mo, by unit load method State the Principle of Virtual work Sketch the Williot s Diagram for the truss in fig. to find B. 6 1 What is the strain energy stored in a rod of length l and axial 7 rigidity AE to an axial force P? 13 Define Virtual work Explain the procedure involved in the deflection of pin jointed plane frames. In the truss shown in fig. no load acts. The member AB gets 4mm too short. The cross sectional area of each member is A = 300 mm and E = 00 GPa. Determine the vertical displacement of joint C. Using the method of virtual work, determine the vertical displacement of point B of the beam shown in fig. Take E = x 10 5 MPa and I = 85x 10 7 mm 4. Table shows the lengths and deformations of the members of the cantilever truss, shown in fig. Construct a Williot diagram and tabulate the displacement of nodes

2 S.NO 16 MARKS PAGE NO Determine the vertical displacement of joint C of the steel truss 1 shown in fig. The cross sectional area of each member is A = 400 mm and E = *10 5 N/mm. Using the principle of virtual work, determine the vertical and horizontal deflection components of joint C of the truss in fig. A = 150*10-6 m and E = 00*10 6 kn/m Determine the vertical and horizontal displacements of the point 3 C of the pin-jointed frame shown in fig. The cross sectional area of AB is 100 sqmm and of AC and BC 150 mm each. E= x 10 5 N/mm. (By unit load method) 4 Using the principle of least work, analyze the portal frame shown in Fig. Using the method of virtual work, determine the horizontal 5 displacement of support D of the frame shown in fig. The values of I are indicated along the members. Take E = 00 x 10 6 KN/m and I = 300 x 10-6 m 4. Using the method of virtual work, determine the horizontal 6 displacement of support D of the frame shown in fig. The values of I are indicated along the members. Take E = 00 x 10 6 KN/m and I = 300 x 10-6 m 4. Using the method of virtual work, determine the horizontal 7 displacement of support D of the frame shown in fig. The values of I are indicated along the members. Take E = 00 x 10 6 KN/m and I = 300 x 10-6 m

3 3 UNIT I TWO MARKS QUESTIONS AND ANSWERS 1. Why it is necessary to compute deflections in structures? Computation of deflection of structures is necessary for the following reasons: (i) If the deflection of a structure is more than the permissible, the structure will not look aesthetic and will cause psychological upsetting of thje occupants. (ii) Excessive deflection may cause cracking in the materials attached to the structure. For example, if the deflection of a floor beam is excessive, the floor finishes and partition walls supported on the beam may get cracked and unserviceable.. What is meant by cambering technique, in structures? Cambering is a technique applied on site, in which a slight upward curve is made in the structure / beams during construction, so that it will straighten out and attain the straight shape during loading. This will considerably reduce the downward deflection that may occur at later stages. 3. Name any four methods used for the computation of deflections in structures. (i) Virtual work method Dummy unit load method (ii) Strain energy method (iii) Willot Mohr diagram method (iv) Method of elastic weights 4. State the difference between strain energy method and unit load method in the determination of deflection of structures. In strain energy method, an imaginary load P is applied at the point where the deflection is desired to be determined. P is equated to Zero in the final step and the deflection is obtained. In unit load method, an unit load (instead of P) is applied at the point where the deflection is desired. 5. What are the assumptions made in the unit load method? Assumptions made in unit load method are 1. The external and internal forces are in equilibrium. Supports are rigid and no movement is possible. 3. The material is strained well within elastic limit. 6. Give the equation that is used for the determination of deflection at a given point i in beams and frames. Deflection at a point i is given by, l M xmxdx i 0

4 Where Mx = moment at a section X due to the applied loads Mx = moment at a section X due to unit load applied at the point i and in the direction of the desired dicplacement = flexural rigidity 11. State the Principle of Virtual work. It states that the work done on a structure by external loads is equal to the internal energy stored in a structure (Ue = Ui) Work of external loads = work of internal loads 4 1. What is the strain energy stored in a rod of length l and axial rigidity AE to an axial force P? Strain energy stored P L U= AE 13. Define Virtual work. The term virtual work means the work done by a real force acting through a virtual displacement or a virtual force acting through a real displacement. The virtual work is not a real quantity but an imaginary one. 14. Explain the procedure involved in the deflection of pin jointed plane frames. 1. Virtual forces k: Remove all the real loads from the truss. Place a unit load on the truss at the joint and in the direction of the desired displacement. Use the method of joints or the method of sections and calculate the internal forces k in each member of the truss.. Real forces F: These forces arre caused only by the real loads acting on the truss. Use the method of sections or the method of joints to determine the forces F in each member. 3. Virtual work equation: Apply the equation of virtual work, to determine the desired displacement. 15. In the truss shown in fig. no load acts. The member AB gets 4mm too short. The cross sectional area of each member is A = 300 mm and E = 00 GPa. Determine the vertical displacement of joint C.

5 5 Solution: Virtual forces, k: Since the vertical displacement of joint C is required, a vertical force of 1 kn is applied at C. The force k in each member is determined as below: By symmetry, RA = RB = ½ Joint A: V = 0 gives KAC cos 36º 5 + ½ = 0 KAC = 0.65 kn (comp) H = 0 gives kab + kac cos 53º 08 = 0 kab = kn (tensile) Joint B: V = 0 gives KBC cos 36º 5 + ½ = 0 KBC = 0.65 kn (comp) Member AB undergoes a deformation, L = m = (k. L) ( C)V = (0.375) (-0.004) = m = -1.5 mm The negative sign indicates that joint C displaced upward, i.e. opposite to the 1 kn vertical load. 16. Using the method of virtual work, determine the vertical displacement of point B of the beam shown in fig. Take E = x 10 5 MPa and I = 85x 10 7 mm 4. Solution: l mmdx = 0 Virtual moment, m. Remove the external load. Apply a unit vertical load at B. Consider a section XX at a distance x from B. m = -1 * x (Hogging moment) Real moment, M. Using the same x co-ordinate, the internal moment (due to the given loading) M is formulated as, M = -5. *.x/ M = -1.5x

6 Virtual work equation. l mmdx ( B)V = 0 CE6501 STRUCTURAL ANALYSIS 1 = 0 ( 1* x)( 1.5x ) dx x x 1.5x = x4 x4 0 = m ( or) 39.3 mm 8 5 *10 x85x10 Hence the vertical displacement of point B = 39.3 mm 3 dx Table shows the lengths and deformations of the members of the cantilever truss, shown in fig. Construct a Williot diagram and tabulate the displacement of nodes. Member Length (mm) Elongation (mm) AC AD BD DC Solution: Fig. shows the Willot s diagram of displacements. Table shows the displacements. Node Displacements (mm) X Y C D

7 SIXTEEN MARKS QUESTIONS AND ANSWERS 7 1. Determine the vertical displacement of joint C of the steel truss shown in fig. The cross sectional area of each member is A = 400 mm and E = *10 5 N/mm. Solution: kfl = AE Virtual forces k. Remove all the (external) loads and apply a unit vertical force at joint C of the truss. Analyze the truss using the method of joints. Take moments about D, VA x 9-1 x3 =0 VA x 9 = 1*3, VA = 1/3 kn VD = Total load VA = 1-1/3 = /3 kn Joint A: Initially assume all forces to be tensile. V 0 gives k cos 45 1/ 3 0 k k k AF AF AF AF cos 45 1/ kn 3cos kn comp H 0 gives k cos 45 k ( 0.471) cos 45 k k AF AB kn AB 0 AB ( tensile ) 0 Joint F : H 0 gives kfa cos 45º + kfe = 0 kfa cos 45º + kfe = - kfa cos 45º = *cos45º= kn kfe = 0.333kN (comp) V 0 kfa cos 45º - kfb = 0 gives

8 8 kfa cos 45º = kfb cos 45º = kfb kfb = kn (tensile) Joint B: V 0 gives Joint C: V 0 gives KBE cos 45º + kfb = 0 k BF k BE cos 45 cos 45 k kn ( comp) BE H 0 gives KCE 1 = 0 KCE + I kn (tensile) H 0 gives kbc kba + kbe cos 45º = 0 kcd - kcb = 0 kcd = kcb = kcd = kn (tensile) Joint D: H 0 gives kde cos45º + kdc = 0 kde = - kdc / cos45º = / cos 45º = kde = 0.94 kn (comp) kbc = kba - kbe cos 45º = (-0.471) cos 45º kbc = kn (tensile) Real Forces F: The real forces in the members due to the given system of external loads are calculated using the method of joints.

9 9 By symmetry RA = RD = Total Load 50 kn Joint A: Initially assume all forces to be tensile. V 0 gives F cos F AF AF 50 / cos kn F AF kn comp H 0 gives F cos 45 F 0 ( 70.71) cos 45 F F AF AB AB AB 50 kn( tensile ) 0 Joint F : H 0 gives FFA cos 45º + FFE = 0 FFA cos 45º + FFE = - FFA cos 45º = *cos45º= -50 kn FFE = -50 kn (comp) V 0 FFA cos 45º - FFB = 0 gives FFA cos 45º = FFB cos 45º = FFB FFB = 50 kn (tensile) Joint B: V 0 gives FBE cos 45º + FFB -50 = 0 FBE cos 45º = - FBF + 50 = = 0 FBE = 0S H 0 gives FBC FBA + FBE cos 45º = 0

10 Joint C: V 0 gives CE6501 STRUCTURAL ANALYSIS FCE 50 = 0 FCE = 50 kn (tensile) FBC = 0 FBC = 50 kn (tensile) 10 H 0 gives FCD - FCB = 0 FCD = FCB = 50 FCD = 50 kn (tensile) Joint D: H 0 gives FDE cos45º + FDC = 0 FDE = - FDC / cos45º = -50/ cos 45º = kn FDE = kn (comp) Virtual Work Equation: kfl = AE AF = BE =DE = m S.No Member k F (kn) L (m) kfl (kn m) 1 AF FE ED DC CB BA FB BE EC kfl kfl 93.58*1000 Nmm kfl 93.58*1000 ( C ) V = mm AE 5 400**10 Vertical displacement of joint C = mm (downward). Using the principle of virtual work, determine the vertical and horizontal deflection components of joint C of the truss in fig. A = 150*10-6 m and E = 00*10 6 kn/m

11 11 Solution: truss. kfl = AE Virtual Forces, kh (for horizontal displacement at C) Remove the real (external load) and apply a unit horizontal force at C of the H 0 gives, HA = 1 kn (CC 1 )/x = tan 50º CC 1 = x tan 50º = (3-x) tan 40º X = (3-x) tan 40º / tan 50º X = x X = 1.39 m (3-x) = = m CC 1 = 1.39*tan 50º = m AC = BC = (1.39) (1.761) (1.477) (1.477) Taking moment about A, VB * 3-1 * = 0 VB = 0.49 kn VA = 0.49 kn Joint A: V 0 gives KAC cos40º = 0 KAC = 0.49/cos40º = 0.64 kn (tensile) m m H 0 gives Joint C: kac cos50º + kab 1 = cos 50º + kab -1 = 0 kab = kn(tensile) V 0 gives kca cos 40º + kcb cos 50º = 0 kcb = kn (comp) Virtual forces kv : (for vertical displacement at C) Remove the real (external load) and apply a unit vertical force at C of the truss. Taking moment about B, VA *3-1*1.761 = 0 VA = kn

12 V B = = kn CE6501 STRUCTURAL ANALYSIS 1 Joint A: V 0 gives KAC cos40º = 0 KAC = /cos40º = kn (comp) H 0 gives kac cos50º + kab = 0 kab = 0.49 kn(tensile) Joint C: V 0 gives cos 40º kcb cos 50º = 0 kcb = kn (comp) Real Forces F: The real forces in the members due to the given external load are calculated as below: The only given force is of magnitude 150 kn and applied at C vertically. Therefore the forces in the members will be 150 times the kv values. FAB = 0.49 * 150 = 73.8 kn (tensile) FAC = * 150 = kn (comp) FCB = * 150 = kn (comp) Virtual Work Equation: kfl = AE S.No Member kh KV F (kn) L(m) FkHL FkVL 1 AB AC CB k FL Horizontal deflection of point C = ( C)h = h AE = / (00 * 10 6 * 150 * 10-6 ) = m = 5.4 mm k FL Vertical deflection of point C = ( C)v = v AE = / (00 * 10 6 * 150 * 10-6 ) = 0.14 m = 14 mm

13 13 3. Determine the vertical and horizontal displacements of the point C of the pinjointed frame shown in fig. The cross sectional area of AB is 100 sqmm and of AC and BC 150 mm each. E= x 10 5 N/mm. (By unit load method) Sol: The vertical and horizontal deflections of the joint C are given by PuL V AE A) Stresses due to External Loading: AC = 3 4 5m Reaction: RA = -3/4 RB = 3/4 Sin θ = 3/5 = 0.6; Cos θ = 4/5 = 0.8 Resolving vertically at the joint C, we get 6 = PAC cos θ + PBC sin θ Resolving horizontally at the joint C, we get PAC cos θ = PBC sin θ; PAC sin θ + PBC sin θ = 6 PAC sin θ = 6 Pu' L AE H PAC = PBC PAC = 6/sin θ = 6/ x 0.6 = 5 KN (tension) PAC = PBC = 5 KN (tension) Resolving horizontally at the joint C, we get PAB = PAC cos θ PAB = 5 cos θ ; PAB = 5 x 0.8 PAB = 4 KN (comp) B) Stresses due to unit vertical load at C: Apply unit vertical load at C. The Stresses in each member will be 1/6 than of those obtained due to external load. u u 5/ 6 u AC AB BC 4 / 6 / 3 C) Stresses due to unit horizontal load at C: Assume the horizontal load towards left as shown in fig. Resolving vertically at the joint C, we get u ' sin u ' sin CA CB

14 14 u ' u ' CA Resolving horizontally at the joint C, we get CB u CB 'cos u CA u 'cos u u 'cos ucb ' 5/8KN( tension) cos x0.8 u ' 5/8KN u CB CA CB CA CB ' 5/8KN( comp) 'cos 1 'cos 1 Resolving horizontally at the joint B, we get u ' u 'cos u u AB AB AB BC ' 5/8x KN ' 0.5KN( comp) Member Length(L) Area P(KN) U (kn) PUL/A U (KN) PU L/A mm (mm) AB /3 640/3-1/ 160 BC /6 500/18 5/8 500/4 CA /6 500/18-5/8 500/4 E = X 10 5 n/mm = 00 KN/m Pul 491 v.45mm AE 00 pu' l 160 h 0. 8mm AE Using the principle of least work, analyze the portal frame shown in Fig. Sol: The support is hinged. Since there are two equations at each supports. They are HA, VA, HD, and VD. The available equilibrium equation is three. (i.e.)m 0, H 0, V 0. The structure is statically indeterminate to first degree. Let us treat the horizontal H ( ) at A as redundant. The horizontal reaction at D will evidently be = (3-H) ( ). By taking moments at D, we get (VA x 3) + H (3-) + (3 x 1) ( 1.5) (6 x ) = 0 VA = 3.5 H/3 VD = 6 VA =.5 + H/3

15 By the theorem of minimum strain energy, U 0 H U AB U BE U CE U DC 0 H H H H 15 (1)For member AB: Taking A as the origin. 1. x M H. x M x H U H AB M M dx H x Hx x H 10.1 Hx x 3 dx () For the member BE: Taking B as the origin. H x 3 3 x M 3.5 Hx M 3H x 3 M x 3 H 3 U BE 1 M M dx H H 1 0 H x 3 Hx x H x dx Hx H x Hx Hx x x dx

16 Hx H x Hx x dx 1 9Hx 13.5x 6x Hx 0.389x 3 3 Hx H H H H 7.9 (3) For the member CE: Taking C as the origin H M (3 H ) x (.5 ) x 3 3 Hx M 6 H.5x 3 U CE H 1 0 M M H = 1 Hx 6 H.5x x Hx H x Hx x Hx x Hx H x Hx x x dx dx 1 = (10.96H ) (4) For the member DC: Taking D as the origin M 3 H x 3x Hx M x x

17 17 U DC H 1 0 M M H dx 1 x 3 0 Hx xdx 1 0 3x Hx dx x Hx 1 3 Hx dx 3 3 x dx = (.67H -8) Subs the values U 0 H 1/ (9-10.) + (8.04H-7.9) + (10.96H-15.78) + (-8+.67H) = H = H = 1.36 KN Hence VA = H/3 = /3 = 3.05 KN VD =.5 + H/3 = /3 =.95 KN MA= MD =0 MB = (-1 x 3 )/ + (1.36 x 3) = -0.4 KN m MC = - (3-H) = - (3-1.36) =-3.8KNm 5. A simply supported beam of span 6m is subjected to a concentrated load of 45 KN at m from the left support. Calculate the deflection under the load point. Take E = 00 x 10 6 KN/m and I = 14 x 10-6 m 4. Solution: Taking moments about B. c VA x 6 45 x 4=0 VA x = 0 VA = 30 KN VB = Total Load VA = 15 KN Virtual work equation: V L 0 mmdx Apply unit vertical load at c instead of 45 KN RA x 6-1 x 4 =0

18 RA = /3 KN RB = Total load RA = 1/3 KN 18 Virtual Moment: Consider section between AC M1 = /3 X1 [limit 0 to ] Section between CB M = /3 X-1 (X- ) [limit to 6 ] Real Moment: The internal moment due to given loading M1= 30 x X1 M = 30 x X -45 (X -) c V 0 m M dx 6 mm dx x x 0 30x1 6 x x 30x 45x 3 dx x 3 x 30x 45x 90 dx dx x 1 90dx 3 0x 15x 0 0x 5x 30x 30x 0 1 0x dx 3 0 5x x 3 180x 6

19 m x10 x14x = The deflection under the load = 57.1 mm ( or) 57.1 mm 6. Using the method of virtual work, determine the horizontal displacement of support D of the frame shown in fig. The values of I are indicated along the members. Take E = 00 x 10 6 KN/m and I = 300 x 10-6 m 4. Solution: l = 0 m M dx Virtual moments, m. Remove the external load and apply a unit load in the horizontal direction at d. the support reactions and internal virtual moments are computed as under.

20 0 (Sign for moments: Left clockwise +ve: Right clockwise +ve) m1 = 1.x1 limits 0 to m m = 1.(+x ) limits 0 to m m3 = 1.x3 limits 0 to 4 m m4 = 1X4 limits 0 to 5 m Real moments, M. Due to the given loading, the support reactions and real moments are computed as under. H 0 gives HA = 50 kn ( ) Taking moments about A, VD x 5 50 x = 0 VD = 50 x /5 = 0 kn ( ) V 0 gives VA = 0 kn ( ) M 1 = 50 X x1 M = 50 X ( + x) 50 X x = 100 knm M 3 = 0 M 4 = 0 X x4 Virtual Work Equation: mmdx ( D)h = 4 (1. x1 )(50x1 ) dx1 1( x )(100) dx (1* x3)* 0dx Ei (1* 4)(0* x 4 ) dx 4 3

21 1 3 1 x 1 00x 100x 40x = = 1/ ( ) = / =133.33/ 00*10 6 *300 *10-6 = m = 0.56 mm Horizontal displacement of support D, ( D)h = 0.56 mm ( ) Using the method of virtual work, determine the horizontal displacement of support D of the frame shown in fig. The values of I are indicated along the members. Take E = 00 x 10 6 KN/m and I = 300 x 10-6 m 4. Solution: l = mmdx 0 Virtual moments, m. Remove the external load and apply unit horizontal load at C. The support reactions and internal virtual moments are computed as shown. H 0 gives HA= 1 kn ( ) Taking moments about C, VA *4 + 1* 5 = 0 4VA = -5 VA = -5/4 = V 0 gives VC = 1.5 kn ( ) i.e. VA = 1.5 kn ( ) M1 = 1.x1 (x1 varies from 0 to.5 m) M = 1.x (x varies from.5 to 5.0 m) M3 = 1.5x3(x3 varies from 0 to 4 m) Real moments. Due to the given external loading, the support react ions and real moments are computed as shown below. H 0 gives HA= 30 kn ( ) Taking moments about A, RA *4-10* 4 * 4/ 30 *.5 = 0 4RC = = 155

22 RC = kn RA = Total load RC = 10* = 1.5 kn M1 = = 30*x1 (x1 varies from 0 to.5) M = 30x 30(x.5) (x varies from.5 to 5) Virtual Work Equation: mmdx ( C)h =.5 0 (1. x1 )(30x1 ) dx1 (1. x ) 30x 30( x.5) dx 4 0 (1.5x3)(38.75x3 5x3 ) dx x 1 = x x x = x4 x4 ( C)h = m *10 *4*10 mm Horizontal displacement of point C = 108 mm UNIT II MOVING LOADS AND INFLUENCE LINES (Determinate and Indeterminate structures) Influence lines for reactions in statically determinate structures influence lines for members forces in pin-jointed frames Influence lines for shear force and bending moment in beam sections Calculation of critical stress resultants due to concentrated and distributed moving loads. Muller Breslaur s principle Influence lines for continuous beams and single storey rigid frames Indirect model analysis for influence lines of indeterminate structures Beggs deformeter. S.NO MARKS PAGE NO 1 What is meant by influence lines? 4

23 3 Draw the influence line diagram for shear force at a point X in a Simply supported beam AB of span l m. 3 Draw the ILD for bending moment at any section X of a simply 4 supported beam and mark the ordinates 4 What are the uses of influence line diagram? 4 5 Sketch the ILDs for the support reactions in a simply supported beam. 6 A simply supported beam of span 10m carries a udl of 0 kn/m over its central 4m length. With the help of influence line diagram, find the shear force at 3m from the left support. 4 7 Sketch the influence line for shear force at a section D in the cantilever shown in fig. 5 8 Draw the influence line diagram for shear force at C and hence determine shear force at C for the given loading.(april/may 04) 5 9 State Muller Bresalu principle Using Muller Bresalu principle, construct the influence line diagram for RB in the continuous beam shown in fig. Sketch the influence line diagram for horizontal reaction at the left support of a single bay single storey portal with fixed legs. 1 Sketch the influence line for reaction at B in the overhanging beam shown in fig.(april/may 05) For the two member determine bent in fig. sketch the influence 13 line for VA. 14 State the principle on which indirect model analysis is based? (Nov/Dec 05) 7 15 What is Begg s deformeter? 7 16 What is dummy length in models tested with Begg s deformeter What are three types of connections possible with the model used with Begg s deformeter What is the use of a micrometer microscope in model analysis with Begg s deformeter? 19 Sketch the plugs used for including axial displacements in the model analysis using Begg s deformeter. 0 Sketch the shear plugs and moment plugs used in Begg s deformeter. 1 Name the types of rolling loads for which the absolute maximum bending moment occurs at the mid span of a beam. What is meant by absolute maximum bending moment in a beam? 3 The portal frame in fig. is hinged at D and is on rollers at A. Sketch the influence line for bending moment at B

24 4 S.NO 16 MARKS PAGE NO A single rolling load of 100 kn moves on a girder of span 0m. (a) Construct the influence lines for (i) shear force and (ii) bending moment for a section 5m from the left support. (b) Construct the influence lines for points at which the maximum shears and maximum bending moment develop. Determine these values. Draw the ILD for shear force and bending moment for a section at 5m from the left hand support of a simply supported beam, 0m long. Hence, calculate the maximum bending moment and shear force at the section, due to a uniformly distributed rolling load of length 8m and intensity 10 kn/m run.(apr/may 05) Two point loads of 100 kn and 00 kn spaced 3m apart cross a girder of span 15m from left to right with the 100 kn load loading. Draw the influence line for shear force and bending moment and find the value of maximum shear force and bending moment at a section, 6m from the left hand support. Also, find the absolute maximum moment due to the given load system A train of 5 wheel loads crosses a simply supported beam of span.5 m. Using influence lines, calculate the maximum positive and negative shear forces at mid span and absolute maximum bending moment anywhere in the span.(nov/dec 05) A girder a span of 18mis simply supported at the ends. It is traversed by a train of loads as shown in fig. The 50 kn load loading. Find the maximum bending moment which can occur (i) under the 00 kn load (ii) Under 50 kn load, using influence line diagrams. Draw the I.L for reaction at B and for the support moment MA at A for the propped cantilever in fig. Compute the I.L ordinates at 1.5 m intervals. In a simply supported girder AB of span om, determine the maximum bending moment and maximum shear force at a section 5m from A, due to the passage of a uniformly distributed load of intensity 0 kn/m, longer than the span. UNIT II TWO MARKS QUESTIONS AND ANSWERS 1. What is meant by influence lines? An influence line is a graph showing, for any given beam frame or truss, the variation of any force or displacement quantity (such as shear force, bending moment, tension, deflection) for all positions of a moving unit load as it crosses the structure from one end to the other.. Draw the influence line diagram for shear force at a point X in a Simply supported beam AB of span l m

25 5 3. Draw the ILD for bending moment at any section X of a simply supported beam and mark the ordinates. 4. What are the uses of influence line diagram? (i) Influence lines are very useful in the quick determination of reactions, shear force, bending moment or similar functions at a given section under any given system of moving loads and (ii) Influence lines are useful in determining the load position to cause maximum value of a given function in a structure on which load positions can vary. 5. Sketch the ILDs for the support reactions in a simply supported beam.

26 6 6. A simply supported beam of span 10m carries a udl of 0 kn/m over its central 4m length. With the help of influence line diagram, find the shear force at 3m from the left support. x l ( l x) l Shear force at X = intensity of udl x area of udl below the udl ( ) = 0* * 4 40 kn 7. Sketch the influence line for shear force at a section D in the cantilever shown in fig.

27 7 8. Draw the influence line diagram for shear force at C and hence determine shear force at C for the given loading. (April/may 04) 9. State Muller Bresalu principle. Muller Bresalu principle states that, if we want to sketch the influence line for any force quantity (like thrust, shear, reaction, support moment or bending moment) in a structure, (i) (ii) we remove from the structure the restraint to that force quantity we apply on the remaining structure a unit displacement corresponding to that forces quantity. 10. Using Muller Bresalu principle, construct the influence line diagram for RB in the continuous beam shown in fig. To construct the influence line for RB, first remove support B. On this reduced structure (now it is a SS beam of span l), apply sufficient force in the RB direction to cause a unit displacement at B. This is actually the case of a SS beam of span l with a central load. The deflection diagram is the ILD for RB.

28 8 11. Sketch the influence line diagram for horizontal reaction at the left support of a single bay single storey portal with fixed legs. To draw the influence line diagram for HA, place support A on rollers so that horizontal displacement at A is permitted. Apply a unit horizontal displacement at A. The resulting displaced shapes are the ILD for HA. 1. Sketch the influence line for reaction at B in the overhanging beam shown in fig.(april/may 05)

29 9 13. For the two member determine bent in fig. sketch the influence line for VA. If we deftly apply Muller Breslau to the problem, we can first remove support A and push A up by unit distance. Since the angle at B will remain unchanged, our unit displacement will result in a rigid body rotation of θ (=1/4 radian) about C. So the column CB will also have a horizontal displacement. (For the part AB, the diagram is just the influence line diagram for shear in a S.S beam) The diagram for BC must be understood to be the influence of horizontal loads on the column on RA. 14. State the principle on which indirect model analysis is based? (Nov/Dec 05) The indirect model analysis is based on the Muller Bresalu principle. Muller Breslau principle has lead to simple method of using models of structures to get the influence lines for force quantities like bending moments, support moments, reactions, internal shears, thrusts, etc. To get the influence line for any force quantity (i) remove the restraint due to the force, (ii) apply a unit displacement in the direction of the force. 15. What is Begg s deformeter? Begg s deformeter is a device to carry out indirect model analysis on structures. It has the facility to apply displacement corresponding to moment, shear or thrust at any desired point in the model. In addition, it provides facility to measure accurately the consequent displacements all over the model. 16. What is dummy length in models tested with Begg s deformeter. Dummy length is the additional length (of about 10 to 1 mm) left at the extremities of the model to enable any desired connection to be made with the gauges. 17. What are three types of connections possible with the model used with Begg s deformeter. (i) Hinged connection

30 (ii) (iii) CE6501 STRUCTURAL ANALYSIS Fixed connection Floating connection What is the use of a micrometer microscope in model analysis with Begg s deformeter? Micrometer microscope is an instrument used to measure the displacements of any point in the x and y directions of a model during tests with Begg s deformeter. 19. Sketch the plugs used for including axial displacements in the model analysis using Begg s deformeter. 0. Sketch the shear plugs and moment plugs used in Begg s deformeter. 1. Name the types of rolling loads for which the absolute maximum bending moment occurs at the mid span of a beam. Types of rolling loads: (i) Single concentrated load (ii) Udl longer than the span (iii) Udl shorter than the span. What is meant by absolute maximum bending moment in a beam? When a given load system moves from one end to the other end of a girder, depending upon the position of the load, there will be a maximum bending moment for every section. The maximum of these maximum bending moments will usually occur near or at the mid span. This maximum of maximum bending moment is called the absolute maximum bending moment, Mmaxmax. 3. The portal frame in fig. is hinged at D and is on rollers at A. Sketch the influence line for bending moment at B.

31 31 To get the influence line diagram for MB, we shall introduce a hinge at B (and remove the resistance to bending moment). Now we get a unit rotation between BA and BC at B. BC cannot rotate since column CD will prevent the rotation. BA would rotate freely (with zero moment). For θ =1 at B, displacement at A = 3m. The displaced position shows the influence line for MB as shown in fig. SIXTEEN MARKS QUESTIONS AND ANSWERS 1. A single rolling load of 100 kn moves on a girder of span 0m. (a) Construct the influence lines for (i) shear force and (ii) bending moment for a section 5m from the left support. (b) Construct the influence lines for points at which the maximum shears and maximum bending moment develop. Determine these values. Solution: (a) To find maximum shear force and bending moment at 5m from the left support: For the ILD for shear,

32 3 l x 0 5 IL ordinate to the right of D = l 0 For the IL for bending moment, IL ordinate at D = x( l x) 5* m l 0 (i) Maximum positive shear force By inspection of the ILD for shear force, it is evident that maximum positive shear force occurs when the load is placed just to the right of D. Maximum positive shear force = load * ordinate = 100 * 7.5 At D, SFmax + = 75 kn. (ii) Maximum negative shear force Maximum negative shear force occurs when the load is placed just to the left D. Maximum negative shear force = load * ordinate = 100 * 0.5 At D, SFmax = -5 kn. (iii) Maximum bending moment Maximum bending moment occurs when the load is placed on the section D itself. Maximum bending moment = load * ordinate = 100 * 3.75 = 375 knm (b) Maximum positive shear force will occur at A. Maximum negative shear force will occur at B. Maximum bending moment will occur at mid span. The ILs are sketched in fig.

33 33 (i) Positive shear force Maximum positive shear force occurs when the load is placed at A. Maximum positive shear force = load * ordinate = 100*1 SFmaxmax + = 100 kn (ii) Negative shear force Maximum negative shear force occurs when the load is placed at B. Maximum negative shear force = load * ordinate = 100 * (-1) SFmaxmax = kn (iii) Maximum bending moment Maximum bending moment occurs when the load is at mid span Maximum bending moment = load * ordinate = 100 * 5 = 500 knm. Draw the ILD for shear force and bending moment for a section at 5m from the left hand support of a simply supported beam, 0m long. Hence, calculate the maximum bending moment and shear force at the section, due to a uniformly distributed rolling load of length 8m and intensity 10 kn/m run.(apr/may 05) Solution:

34 34 (a) Maximum bending moment: Maximum bending moment at a D due to a udl shorter than the span occurs when the section divides the load in the same ratio as it divides the span. A1 D AD In the above fig. 0.5, A1 D M, B1D 6M B1D BD Ordinates: Ordinate under A1 = (3.75/5)*3 =.5 Ordinate under B1 = (3.75/15)*9 =.5 Maximum bending moment = Intensity of load * Area of ILd under the load (3.75.5) *8 = 10 * At D, Mmax = 40 knm (b) Maximum positive shear force Maximum positive shear force occurs when the tail of the UDL is at D as it traverses from left to right.

35 Ordinate under B1 = * (15 8) Maximum positive shear force = Intensity of load * Area of ILD under load ( ) *8 = 10* SFmax = + 44 knm (c) Maximum negative shear force Maximum negative shear force occurs when the head of the UDL is at D as it traverses from left to right. Maximum negative shear force = Intensity of load * Area of ILD under the load = 10(1/*0.5*5) Negative SFmax = 6.5 kn. 3. Two point loads of 100 kn and 00 kn spaced 3m apart cross a girder of span 15m from left to right with the 100 kn load loading. Draw the influence line for shear force and bending moment and find the value of maximum shear force and bending moment at a section, 6m from the left hand support. Also, find the absolute maximum moment due to the given load system. Solution: (a) Maximum bending moment

36 36 x( l x) 6 *9 Max. Ordinate of ILD = 3.6 m l 15 Maximum bending moment at D occurs under the critical load. This load, when it moves from left to right of D changes the sign of Lr, the differential loading rate, where, Wleft Wright Lr x l x Now, let us try with 00 kn load. Firstly, keep this 00 kn load to the left of D. L r Wleft Wright =. ( ve) x l x 6 9 Moving this load to the right of D, Wleft Wright Lr = ( ve) x l x 6 9 Since, the sign of Lr changes, the 00 kn load is critical. The maximum bending moment occurs at D when this 00 kn load is placed at D.

37 Ordinate under 100 kn load = * Maximum bending moment = ( load * ordinate ) 00* *. 4 (b) Maximum shear force: (i) Positive shear force Fig. shows the load position for the absolute maximum positive shera force. If the load train is moved to the left, the positive contribution due to the bigger (160 kn) load is lost and a negative contribution is obtained. If the load train moves to the right, the ordinates under to the loads decrease. Hence, the indicated load position is the critical one. Ordinate under 00 kn load = Ordinate under 100 kn load = Maximum positive shear force = 00 (0.6) (0.4) SFmax = kn. (ii) Negative shear force

38 38 Trying with 100 kn load, first keep this 100 kn load to the left of D. Then move this load to the right of D by 3m. If the value of the shear increment Si is a negative value, it includes a decrease in negative shear force. Wc 300*3 Si W ( ve) l 15 Hence, the negative shear force decreases when this load train is moved to the right of D. Hence, to get maximum shear force, this 100 kn load should be kept just to the left of D. Ordinate under 100 kn load = Ordinate under 00 kn load = * Maximum negative shear force = 100* *0. = 80 = - 80 kn SFmax (c) Absolute maximum bending moment (i) Resultant of the loads Taking moments about 00 kn load, 100*3 = R.x, R = 300 kn. X = 1.0 m Absolute maximum bending moment occurs under the load which is nearer to the resultant R. The critical position is when the resultant R and the load are at equal distance from the centre of span. Distance of this 00 kn from C = Distance of R from c. x( l x) Maximum ordinate of ILD (i.e) ordinate under 00 kn load = m l Ordinate under 100 kn load = (3.733*6)/9 =.489 m Absolute maximum bending moment, Mmaxmax = ( load * Dis tan ce) = 00 * *.489 = knm

39 39 4. A train of 5 wheel loads crosses a simply supported beam of span.5 m. Using influence lines, calculate the maximum positive and negative shear forces at mid span and absolute maximum bending moment anywhere in the span.(nov/dec 05) Solution: (a) Maximum shear force (i) Positive shear force To determine the load position to get the maximum positive shear force, let us keep all the loads to the right of C. Then move W1 load to the left of C by.5 m. if the sign of shear increments Si is negative, it will indicate that W1 shall be retained at C. Wc Si W 1 l W = Total load on the span = = 118 kn. C = Distance through which the load train is moved =.5 m

40 *.5 S i ( ve).5 Since Si is positive, the shear force increases due to thr shifting of W1 to the left of C. Again, let us move W to the left of C by.5 m to check whether the shera force further increases or not. Wc 1180*.5 Si W ( ve) l.5 Since Si is negative, it undicates that to get maximum positive shear force, W should stay just right of C. Ordinate of ILD 0.5 *(11.5.5) 0.39 Ordinate under W1 = Ordinate under W = Ordinate under W3 = * Ordinate under W4 = * Ordinate under W5 = * Maximum positive shear force = ( laodxordinate) (0.17) At C, SFmax + = 30.8 kn. = 10 (-0.39) (0.5) +400 (0.39) +60(0.8) + 40 ii) Negative shear force

41 41 To determine the position of loads to get the maximum negative shear force, move the loads one by one to the right of C and computer the value of Si.If Si becomes negative it will indicate a decrease in negative shear force due to that movement. First let us move the leading W5 to the right of C by.5 m and calculate Si Wd Si= W5 l W5 = 40 kn; d =.5 m 1180x.5 Si = ( ve).5 Since Si is ve it indicates that W5 should stay just to the left of C. Ordinates of ILD: x 11.5 Ordinate under W5 = 0. 5 l Ordinate under W4 = x (11.5.5) Ordinate under W3 = x ( ) Ordinate under W = x ( ) Ordinate under W1 = x ( ) Maximum negative shear force at C = 40(-0.5)+60(-0.39)+(400(-0.78) +160( )

42 +10(-0.056) 4 Fmax = kn. b) Absolute maximum bending moment i) Position of resultant of all loads Taking moments about W1, 10(0)+160(.5)+400(5.0)+60(7.5)+40(10.0) =R. x R = 1180kN x = 5,7m from W1 ii) Location of absolute maximum bending moment Absolute maximum bending moment occurs under the load, which is nearest to the resultant R.(In this problem W3 is nearest to the resultant R ).The distance between C and R and the distance between C and W3 shall be equal. Distance between R and center of span(c) = ½ (0.7) =0.36 m In this fig. Shows the IL for bending moment at the critical spot D, m fro A. Ordinates of ILD:

43 43 Maximum ordinate of ILD x( l x) 10.89( ) (i.e) Ordinate under W3 = 5. 6 l Ordinate under W = x Ordinate under W1 = x Ordinate under W4 = x Ordinate under W5 = x Absolute maximum bending moment =10 (3.04)+160(4.33)+400(5.6)+60(4.41)+ 40 (3.) Mmax max = 50. kn.m 5) A girder a span of 18mis simply supported at the ends. It is traversed by a train of loads as shown in fig. The 50 kn load loading. Find the maximum bending moment which can occur (i) under the 00 kn load (ii) Under 50 kn load, using influence line diagrams.

44 Solution: CE6501 STRUCTURAL ANALYSIS 44 a) Maximum bending moment i) Under 00 kn loads. To get the maximum bending moment under W3 the resultant R and W3 should be at equal distances from the center of the span C.For that the point of action of resultant R should be determined first. a) Resultant of loads: R = 450 kn Taking moments about W4 00(3) +100 (3+) +50(3 ++3) = 450 x x = 3.33 from W4 b) Bending moment under 00 kn load Distance between C and 00 kn load = Distance between C and R = 0.33/ = m.

45 45 Ordiantes of ILD : ILD under W3 = x ( l x) l X= 8.335m l-x = = m ILD under W3 = 8.835* m ILD under W4 = 4.5* m ILD under W = 4.5*( ) m ILD under W1 = 4.5* m Bending moment under the 00 kn load = 00 (4.5) (.97) (3.5) +50 (.05) = knm. (ii) Bending moment under the load 50 kn load To get the maximum bending moment under W1, W and R must be at equal distances from the centre of span. Distance between C and R = Distance between C and W1 = ½ ( ) = ½ (4.67) =.335 m Ordinates of ILD:

46 46 ILD under W1 = x( l x) l * m ILD under W = x( l x) l 4. * m ILD under W3 = x( l x) l 4. * m ILD under W4 = x( l x) l 4.* m Bending moment under the load 50 kn load m. = 4. (50) (100) +.35 (00) +1.4 (100) = 1113 kn 6.Draw the I.L for reaction at B and for the support moment MA at A for the propped cantilever in fig. Compute the I.L ordinates at 1.5 m intervals. Solution: Remove the restraint due to RB (remove support B) Apply a unit displacement (upward). to unit load applied at B. When RB = 1, then YXB is the displacement at section x due d y Mx = - dx y d RB. x 1. x : dx x

47 47 dy x C1 dx 3 x y C1x C 6 At x= 1, y = 0, dy/dx = 0 1 Hence, C1 = 7, C = -576 YXB = 1 x 6 7x 576 YBB (at x = 0 ) = 576 When we plot this against x, we get the I.L for RB. Xm RB Fig. is the influence line diagram for RB.

48 48 To get the I.L for MA have to (i) (ii) Introduce a hinge at A and We have to apply a unit rotation at A. Instead we will apply a unit moment at A find the general displacement at any x from B. We will then divide the displacement by the actual rotation at A. Due to M R B A 1 R A 1/1 d y x M x dx 1 dy x Ei C1 dx 4

49 49 The ordinates of the I.L.D for MA at 1.5 m intervals are tabulated below. X ILD

50 7. In a simply supported girder AB of span om, determine the maximum bending moment and maximum shear force at a section 5m from A, due to the passage of a uniformly distributed load of intensity 0 kn/m, longer than the span. Solution: (i) Maximum bending moment Since the udl is longer than the, the criterion for maximum bending moment at a section is that the whole span should be loaded as shown in fig. 50 (ii) Maximum shear force Maximum negative shear force at a section occurs when the head of the load reaches the section ((i.e. when the left portion AX is loaded and right portion XB is empty)

51 51 (iii) Maximum positive shear force: Maximum positive shear force occurs at X when the tail of the load is at X as it moves from left to right. (i.e. AX is empty and the portion XB is loaded) Maximum positive shear force = RA UNIT III ARCHES Arches as structural forms Examples of arch structures Types of arches Analysis of Three hinged, two hinged and fixed arches, parabolic and Circular arches Settlement and temperature effects. S.NO MARKS PAGE NO 1 What is an arch? Explain. 4 What is a linear arch? 4 3 State Eddy s theorem. 4 4 What is the degree of static indeterminacy of a three-hinged parabolic arch? A three hinged parabolic arch hinged at the crown and springings 5 has a horizontal span of 1m and a central rise of.5m. it carries a udl of 30 kn/m run over the left hand half of the span. Calculate the resultant at the end hinges. 6 Explain with the aid of a sketch, the normal thrust and radial shear in an arch rib. 7 Explain with the aid of a sketch, the normal thrust and radial shear in an arch rib. 8 Obtain the equation to a parabolic arch (span l and rise r ) with origin at the left hinge. 9 What is the difference between the basic action of an arch and a suspension cable? 10 Under what conditions will the bending moment in an arch be 11 zero throughout? A three-hinged semicircular arch carries a point load of 100 kn at the crown. The radius of the arch is 4m. Find the horizontal reactions at the supports

52 1 A three-hinged semicircular arch of radius 10m carries a udl of kn/m over the span. Determine the horizontal and vertical 8 reactions at the supports. 13 Determine H, VA and VB in the semicircular arch shown in fig Draw the influence line for horizontal thrust in a three hinged parabolic arch having a span of 40m and central rise of 10m Draw the influence line for radial shear at a section of threehinged arch Sketch the influence line diagram for horizontal thrust in a twohinged parabolic arch Distinguish between two hinged and three hinged arches Explain rib shorting in the case of arches Explain the effect of yielding of support in the case of an arch For the arch shown in fig. Find the normal thrust at a section 5m from the left end, given θ at D = 0º A three-hinged parabolic arch has a horizontal span of 36m with a central rise of 6m. A point load of 40 kn moves across the span from the left to the right. What is the absolute maximum 1 positive bending moment that will occur in the arch? A three-hinged parabolic arch of span 30m has a central rise of 5m. Find the increase in the rise of the arch if the temperature rise through 30ºC. Take coefficient of linear expansion of the 1 arch material as 1 * 10-6 /ºC. 3 Determine the support reactions in the arch shown in fig S.NO 16 MARKS PAGE NO A 3 hinged arch of span 40m and rise 8m carries concentrated loads of 00 kn and 150 kn at a distance of 8m and 16m from the left end and an udl of 50 kn/m on the right half of the span. Find the horizontal thrust. A parabolic 3-hinged arch carries a udl of 30kN/m on the left half of the span. It has a span of 16m and central rise of 3m. Determine the resultant reaction at supports. Find the bending moment, normal thrust and radial shear at xx, m from left support. A parabolic 3-hinged arch carries loads as shown in fig. Determine the resultant reactions at supports. Find the bending moment, normal thrust and radial shear at D, 5m from A. What is the maximum bending moment? A 3-hinged arch is circular, 5 m in span with a central rise of 5m. It is loaded with a concentrated load of 10 kn at 7.5m from the left hand hinge. Find the (a) Horizontal thrust (b) Reaction at each end hinge (c) Bending moment under the load A 3-hinged arch is circular, 5 m in span with a central rise of 5m. It is loaded with a concentrated load of 10 kn at 7.5m from the left hand hinge. Find the following

53 6 7 CE6501 STRUCTURAL ANALYSIS A three hinged circular arch of span 16m and rise 4m is subjected to two point loads of 100 kn and 80 kn at the left and right quarter span points respectively. Find the reactions at supports. Find also the bending moment, radial shear and normal thrust at 6m from left support. A symmetrical three hinged parabolic arch of span 40m and rise 8m carries an udl of 30 kn/m over left of the span. The hinges are provided at these supports and at the center of the arch. Calculate the reactions at the supports. Also calculate the bending moment, radial shear, and normal thrust at distance of 10 m from the left support. UNIT II TWO MARKS QUESTIONS AND ANSWERS What is an arch? Explain. An arch is defined as a curved girder, having convexity upwards and supported at its ends. The supports must effectively arrest displacements in the vertical and horizontal directions. Only then there will be arch action.. What is a linear arch? If an arch is to take loads, say W1, W, and W3 and a vector diagram and funicular polygon are plotted as shown; the funicular polygon is known as the linear arch or theoretical arch. The polar distance ot represents the horizontal thrust. The links AC, CD, DE and EB will br under compression and there will be no bending moment. If an arch of this shape ACDEB is provided, there will be no bending moment. 3. State Eddy s theorem.

54 Eddy s theorem states that The bending moment at any section of an arch is proportional to the vertical intercept between the linear arch (or theoretical arch) and the center line of the actual arch. BMx = ordinate O O3 * scale factor What is the degree of static indeterminacy of a three hinged parabolic arch? For a three-hinged parabolic arch, the degree of static indeterminacy is zero. It is statically determinate. 5. A three hinged parabolic arch hinged at the crown and springing has a horizontal span of 1m and a central rise of.5m. it carries a udl of 30 kn/m run over the left hand half of the span. Calculate the resultant at the end hinges.

55 55 6. Explain with the aid of a sketch, the normal thrust and radial shear in an arch rib. Let us take a section X of an arch. Let θ be the inclination of the tangent at X. if H is the horizontal thrust and V the net vertical shear at X, from theb free body of the RHS of the arch, it is clear that V and H will have normal and radial components given by, N = H cos θ + V sin θ R = V cosθ H sin θ 7. Explain with the aid of a sketch, the normal thrust and radial shear in an arch rib. Parabolic arches are preferable to carry distributed loads. Because, both, the shape of the arch and the shape of the bending moment diagram are parabolic. Hence the intercept between the theoretical arch and actual arch is zero everywhere. Hence, the bending moment at every section of the arch will be zero. The arch will be under pure compression that will be economical.