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# Structural analysis 2

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### Structural analysis 2

1. 1. P a g e | 198 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry STRUCTURAL ANALYSIS – 2 UNIT – 1 1. 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. 2. State the general cable theorem. The general cable theorem helps us determine the shape of a cable supported at two ends when it is acted upon by vertical forces. It can be stated as: “At any point on a cable acted upon by vertical loads, the product of the horizontal component of cable tension and the vertical distance from that point to the cable chord equals the moment this would occur at that section if the loads carried by the cable were acting on an simply-supported beam of the same span as that of the cable.” 3. What are the various types of hinges in arch? (or) What are the types of arches according to the support conditions?  Three hinged arch  Two hinged arch  Single hinged arch  Fixed or hinge less arch 4. What are the types of arches according to their shapes?  Curved arch
2. 2. P a g e | 199 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry  Parabolic arch  Elliptical arch  Polygonal arch 5. Define horizontal thrust. In a 3 hinged arch, the force H is calculated by equating the bending moment at the central hinge to zero. The horizontal thrust H reduces the beam bending moment called µx. 𝐴𝑐𝑡𝑢𝑎𝑙𝑙𝑦 𝑖𝑛 𝑎𝑛 𝑎𝑟𝑐ℎ, 𝑀 𝑥 = 𝜇 𝑥 − 𝐻 𝑦 6. What is a linear arch? If an arch is to take loads, say W1, W2, and W3 and a vector diagram and funicular polygon are plotted as shown in figure, the funicular polygon is known as the linear arch or theoretical arch. The polar distance ‘o t’ represents the horizontal thrust. The links AC, CD, DE and EB will be 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. 3. P a g e | 200 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry For a given set of vertical loads W1, W2……etc. we can have any number of linear arches depending on where we chose ‘O’ or how much horizontal thrust (o t) we choose to introduce. 7. Draw the influence line for horizontal reaction, H in a three hinged stiffening girder. 8. Why stiffening girders are necessary in the suspension bridges?  Stiffening girders enable the suspension bridge decks to remain fairly level  Whatever be the live load on the deck slab, the stiffening girders will convert and transmit the load on the deck slab as a uniformly distributed load and thereby help the cable retain the parabolic shape during the passage of loads
4. 4. P a g e | 201 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry  The dead load of the girders which is a UDL is directly transmitted to the cables and is taken up entirely by the tension in the cables  Thus the uniformly distributed dead load will not cause any shear force or bending moment in the stiffening girder  The stiffening girders will have to resist the Shear force and bending moment due to live loads 9. Write the expression for horizontal thrust in a three hinged parabolic arch carrying UDL over entire span. 𝐻 = 𝑤 𝑙2 8𝑦𝑐 10.Write the expression for horizontal thrust of a semicircular arch. 𝐻 = 𝑊 cos2 𝛷 𝜋 If the load is applied at the centre, we get 𝐻 = 𝑊 𝜋 = 0.318 𝑊 11.A flexible cable 20m long is supported at two ends at the same level. The supports are 16m apart. Determine the dip of the cable. GIVEN DATA: S = 20m l = 16m TO FIND: d =?
5. 5. P a g e | 202 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry SOLUTION: 𝑆 = 𝑙 + 8 3 𝑑2 𝑙 𝑑 = √ ( 𝑆 − 𝑙 ) × 3𝑙 8 d = 4.89m 12.State the “Eddy’s theorem” for an arch. Eddy’s theorem states that the bending moment at any section of an arch is equal to the vertical intercept between the linear arch and the center line of the actual arch. 𝐵𝑀 𝑥 = 𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑂2 𝑂3 × 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 13.What is the 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.
6. 6. P a g e | 203 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 14.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 q be the inclination of the tangent at X. If H is the horizontal thrust and V is the vertical shear at X, from the free body of the RHS of the arch, it is clear that V and H will have normal and radial components given by, 𝑅𝑎𝑑𝑖𝑎𝑙 𝑠ℎ𝑒𝑎𝑟 ( 𝑅 𝑥 ) = 𝑉𝑥 cos 𝜃 − 𝐻 sin 𝜃 𝑁𝑜𝑟𝑚𝑎𝑙 𝑡ℎ𝑟𝑢𝑠𝑡 ( 𝑁𝑥 ) = 𝑉𝑥 sin 𝜃 + 𝐻 cos 𝜃 15.Which of the two arches, viz. circular and parabolic is preferable to carry a uniformly distributed load? Why? Parabolic arches are preferably to carry distributed loads. Because, both the shape of the arch and the shape of the bending moment diagram are parabolic. Hence the vertical 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 which will be economical.
7. 7. P a g e | 204 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 16.What is the difference between the basic action of an arch and suspension cable? An arch is essentially a compression member which can also take bending moments and shears. Bending moments and shears will be absent if the arch is parabolic and the loading uniformly distributed. A cable can take only tension. A suspension bridge will therefore have a cable and a stiffening girder. The girder will take the bending moment and shears in the bridge and the cable, only tension. Because of the thrusts in the cables and arches, the bending moments are considerably reduced. If the load on the girder is uniform, the bridge will have only cable tension and no bending moment on the girder. 17.Under what conditions will the bending moment in an arch be zero throughout? The bending moment in an arch throughout the span will be zero, if  The arch is parabolic  The arch carries UDL throughout the span 18.Indicate the positions of a moving point load for maximum negative and positive bending moments in a three hinged arch. Considering a three hinged parabolic arch of span ‘l’ and subjected to a moving point load W, the position of the point load for  Maximum negative bending moment is 0.25l from end supports.  Maximum positive bending moment is 0.211l from end supports.
8. 8. P a g e | 205 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 19.Draw the ILD for bending moment at a section x at a distance x from the left end of a three hinged parabolic arch of span ‘l’ and rise ‘h’. 𝑀 𝑥 = 𝜇 𝑥 − 𝐻 𝑦 20.Distinguish between two hinged and three hinged arches. TWO HINGED ARCHES THREE HINGED ARCHES Statically indeterminate to first degree Statically determinate Might develop temperature stresses Increase in temperature causes increase in central rise. No stresses Structurally more efficient Easy to analysis. But in construction, the central hinge may involve additional expenditure. Will develop stresses due to sinking of supports Since this is determinate, no stresses due to support sinking. 21.Explain rib shortening in the case of arches. In a two hinged arch, the normal thrust which is a compressive force along the axis of the arch will shorten the rib of the arch. This in turn will release part of the horizontal thrust. Normally, this effect is not considered in the analysis (in the case of two hinged arches).
9. 9. P a g e | 206 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry Depending upon the importance of the work we can either take into account or omit the effect of rib shortening. This will be done by considering (or omitting) strain energy due to axial compression along with the strain energy due to bending in evaluating H. 22.What are cable structures? Long span structures subjected to tension and uses suspension cables for supports. Examples of cable structures are suspension bridges, cable stayed roof. 23.Explain the yielding of support in the case of an arch. Yielding of supports has no effect in the case of a 3 hinged arch which is determinate. These displacements must be taken into account when we analyze 2 hinged or fixed arches under 𝜕𝑈 𝜕𝐻 = ∆𝐻 Instead of zero 𝜕𝑈 𝜕𝑉 𝐴 = ∆𝑉𝐴 Instead of zero Here U is the strain energy of the arch and ΔH and ΔVA are the displacements due to yielding of supports. 24.Write the formula to calculate the change in rise in three hinged arch. 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑟𝑖𝑠𝑒 = ( 𝑙2 + 4𝑦𝑐 2 4𝑦𝑐 ) × 𝛼𝑇 Where, l = span length of the arch yc = central rise of the arch α = coefficient of thermal expansion
10. 10. P a g e | 207 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry T = change in temperature 25.In a parabolic arch with two hinges how will you calculate the slope of the arch at any point? 𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 𝑎𝑟𝑐ℎ ( 𝜃 ) = tan−1 ( 4𝑦𝑐 𝑙2 × ( 𝑙 − 2𝑥 )) Where, θ = slope at any point x (or) inclination of tangent at x l = span length of the arch yc = central rise of the arch 26.How will you calculate the horizontal thrust in a two hinged parabolic arch if there is a rise in temperature? 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑡ℎ𝑟𝑢𝑠𝑡 ( 𝐻 ) = 𝛼 𝑇 𝑙 𝐸 𝐼 ∫ 𝑦2 𝑑𝑥 𝑙 0 Where, l = span length of the arch y = rise of the arch at any point x α = coefficient of thermal expansion T = change in temperature E = Young’s Modulus of the material of the arch I = Moment of Inertia 27.What is the true shape of cable structures? Cable structures especially the cable of a suspension bridge is in the form of a catenary. Catenary is the shape assumed by a string / cable freely suspended between two points.
11. 11. P a g e | 208 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 28.What is the nature of force in the cables? Cables of cable structures have only tension and no compression or bending. 29.What is a catenary? Catenary is the shape taken up by a cable or rope freely suspended between two supports and under its own self weight. 30.Mention the different types of cable structures?  Cable over a guide pulley  Cable over a saddle 31.Briefly explain cable over a guide pulley. Cable over a guide pulley has the following properties:  Tension in the suspension cable = tension in the anchor cable  The supporting tower will be subjected to vertical pressure and bending due to net horizontal cable tension 32.Briefly explain cable over saddle. Cable over saddle has the following properties:  Horizontal component of tension in the cable = horizontal component of tension in the anchor cable  The supporting tower will be subjected to only vertical pressure due to cable tension 33.What are the main functions of stiffening girder in suspension bridges?  They help in keeping the cables in shape
12. 12. P a g e | 209 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry  They resist part of the Shear force and bending moment due to live loads 34.What is the degree of indeterminacy of a suspension bridge with two hinged stiffening girder? The two hinged stiffening girder has one degree of indeterminacy. 35.Differentiate between plane truss and space truss. PLANE TRUSS SPACE TRUSS All members lie in one plane This is a three dimensional truss All joints are assumed to be hinged All joints are assumed to be ball and socketed 36.Give some examples of beams curved in plan. Curved beams are found in the following structures.  Beams in the bridge negotiating a curve  Ring beams supporting a water tank  Beams supporting corner lintels  Beams in ramps 37.What are the forces developed in beams curved in plan? Beams curved in plan will have the following forces developed in them.  Bending moments  Shear forces  Torsional moments
13. 13. P a g e | 210 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 38.Define tension coefficient of a truss member. The tension coefficient for a member of a truss is defined as the pull or tension in the member divided by its length (i.e.) the force in the member per unit length. 39.What are the significant features of circular beams on equally spaced supports?  Slope on either side of any support will be zero  Torsional moment on every support will be zero 40.Give the expression for calculating equivalent UDL on a girder. 𝑊𝑒 = 𝑡𝑜𝑡𝑎𝑙 𝑙𝑜𝑎𝑑 𝑜𝑛 𝑔𝑖𝑟𝑑𝑒𝑟 𝑠𝑝𝑎𝑛 𝑜𝑓 𝑔𝑖𝑟𝑑𝑒𝑟 41.Give the expression for determining the tension T in the cable. The tension developed in the cable is given by, 𝑇 = √ 𝐻2 + 𝑉2 Where, H = horizontal component V = vertical component 42.What are cables made of? Cables can be of mild steel, high strength steel, stainless steel, or polyester fibres. Structural cables are made of a series of small strands twisted or bound together to form a much larger cable. Steel cables are either spiral strand, where circular rods are twisted together or locked coil strand,
14. 14. P a g e | 211 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry where individual interlocking steel strands form the cable (often with a spiral strand core). Spiral strand is slightly weaker than locked coil strand. Steel spiral strand cables have a Young's modulus, E of 150 ± 10 kN/mm² and come in sizes from 3 to 90 mm diameter. Spiral strand suffers from construction stretch, where the strands compact when the cable is loaded. 43.Give the horizontal and vertical components of a cable structure subjected to UDL. The horizontal and vertical reactions are given by, 𝐻 = 𝑤 𝑆2 8𝑑 And 𝑉 = 𝑤 𝑆 2 44.What is meant by “Reaction locus” for a two hinged arch? The Reaction locus is a line which gives the point of intersection of the two reactions for any position of an isolated load. 45.Give the range of central dip of a cable. The central dip of a cable ranges from 1/10 to 1/12 of the span.
15. 15. P a g e | 212 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 46.Give the types of significant cable structures.  Linear structure  Suspension bridges  Draped cables  Cable stayed beams or trusses  Cable trusses  Straight tensioned cables  Three dimensional structure  Bicycle wheel roof  3D cable trusses  Tensegrity structures  Tensairity structures UNIT – 2 & 3 1. Where do you get the rolling loads in practice? Shifting of load positions is common enough in buildings. But they are more pronounced in bridges and in gantry girders over which vehicles keep rolling. 2. List the categories of rolling loads on beams.  Single concentrated load  UDL longer than the beam span  UDL shorter than the beam span  Two wheel axles separated by a fixed distance  Multiple wheel axles (train of loads)
17. 17. P a g e | 214 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 6. What is the absolute maximum bending moment due to a UDL longer than the span of a simply supported beam? When a simply supported beam is subjected to a moving UDL longer than the span, the absolute maximum bending moment occurs when the whole span is loaded. 𝑀max 𝑚𝑎𝑥 = 𝑤 𝑙2 8 7. State the location of a maximum shear force in a simple beam with any kind of loading. In a simple beam with any kind of load, the maximum positive Shear force occurs at the left hand support and maximum negative Shear force occurs at right hand support. 8. 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 bending moments will usually occur near or at the mid span. The maximum of maximum bending moments is called the absolute maximum bending moment. 9. What is meant by influence lines? An influence line is a graph showing, for any given 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.
18. 18. P a g e | 215 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry An influence line for any given point or section of structure is a curve whose ordinates represent to scale the variation of a function, such as shear force, bending moment, deflection etc. at the point or section, as the unit load moves across the structure. ILD for determinate beam is linear and for indeterminate structure is curvilinear. 10.What are the uses of influence diagrams?  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  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. 11.State Muller Breslau principle. Muller-Breslau principle states that, if we want to sketch the influence line for any force quantity (like thrust, shear, and reaction, support moment or bending moment) in a structure,  We remove from the structure the restraint to that force quantity and  We apply on the remaining structure a unit displacement corresponding to that force quantity. The resulting displacements in the structure are the influence line ordinates sought. 12.Write the uses of Muller Breslau principle.  It is the most important tool in obtaining influence lines for statically determinate as well as statically indeterminate structures
20. 20. P a g e | 217 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry girder, the diagram that we obtain is the maximum shear force diagram. This diagram yields the ‘design shear’ for each cross section. 16.What do you understand by the term reversal of stresses? In certain long trusses the web members can develop either tension or compression depending upon the position of live loads. This tendency to change the nature of stresses is called reversal of stresses. 17.Draw the ILD for shear force shear force at a point x in a simply supported beam AB of span l. 18.State Maxwell – Betti’s theorem.
21. 21. P a g e | 218 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry In a linearly elastic structure in static equilibrium acted upon by either of two systems of external forces, the virtual work done by the first system of forces in undergoing the displacements caused by the second system of forces is equal to the virtual work done by the second system of forces in undergoing the displacements caused by the first system of forces. Maxwell Betti’s theorem helps us to draw influence lines for structures. 19.Draw the influence line for radial shear at a section of a three hinged arch. 𝑅𝑎𝑑𝑖𝑎𝑙 𝑠ℎ𝑒𝑎𝑟 ( 𝑅 𝑥 ) = 𝑉𝑥 cos 𝜃 − 𝐻 sin 𝜃 Where, θ is the inclination of tangent at x 20.What is the necessity of model analysis?  When the mathematical analysis of problem is virtually impossible  Mathematical analysis though possible is so complicated and time consuming that the model analysis offers a short cut The importance of the problem is such that verification of mathematical analysis by an actual test is essential
22. 22. P a g e | 219 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 21.Draw the ILD for bending moment at any section x of a simply supported beam and mark the ordinates. 22.Sketch the ILD for the normal thrust at a section X of a symmetric three hinged parabolic arch. 𝑁𝑜𝑟𝑚𝑎𝑙 𝑡ℎ𝑟𝑢𝑠𝑡 ( 𝑁𝑥 ) = 𝑉𝑥 sin 𝜃 + 𝐻 cos 𝜃 Where, θ is the inclination of tangent at x 23.Define Maxwell’s reciprocal theorem or Bette’s theorem. The work done by the first system of loads due to displacements caused by a second system of loads equals to the work done by the second system of loads due to displacements caused by the first system of loads.
23. 23. P a g e | 220 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 24.Define similitude. Similitude means similarity between two objects namely the model and the prototype with regard to their physical characteristics.  Geometric similitude is similarity of form  Kinematic similitude is similarity of motion  Dynamic and / or mechanical similitude is similarity of masses and / or forces 25.State the principle on which indirect model analysis is based. The indirect model analysis is based on the Muller Breslau principle. Muller Breslau principle has led to a 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,  Remove the restraint due to the force  Apply a unit displacement in the direction of the force  Plot the resulting displacement diagram This diagram is the influence line for the force. 26.What is the principle of dimensional similarity? Dimensional similarity means geometric similarity of form. This means that all homologous dimensions of prototype and model must be in some constant ratio. 27.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
24. 24. P a g e | 221 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry moment, shear or thrust at any desired points in the model. In addition, it provides facility to measure accurately the consequent displacements all over the model. 28.Name any four model making materials. Perspex, Plexiglas, acrylic, plywood, sheet araldite and Bakelite are some of the model making materials. Micro – concrete, mortar and plaster of Paris can also be used for models. 29.What is dummy length in models tested with Begg’s deformeter? Dummy length is the additional length (of about 10 to 12mm) left at the extremities of the model to enable any desired connection to be made with the gauges. 30.What are the three types of connections possible with the model used with Begg’s deformeter?  Hinged connection  Fixed connection  Floating connection 31.What are the uses of a micrometer microscope in model analysis with Begg’s deformeter? Micrometer microscope is an instrument used to measure the displacement of any point in the x and y directions of a model during tests with Begg’s deformeter.
25. 25. P a g e | 222 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry UNIT – 4 1. What is meant by yield stress? Most structural materials have under gradually increasing strain an elastic and plastic stage. Plastic stage mark the stage at which increased strain does not produce in stress. The stress consequent to stretching stabilize at a value is known as yield stress. 2. What are the basic conditions to be satisfied for plastic analysis?  Mechanism condition The ultimate load or collapse load is reached when a mechanism is formed. There must, however, be just enough plastic hinges that a mechanism is formed.  Equilibrium condition The summation of the forces and moments acting on a structure must be equal to zero.  Plastic moment condition The bending moment anywhere must not exceed the fully plastic moment. 3. What are the basic conditions to be satisfied for elastic analysis?  Continuity equation  Equilibrium condition
26. 26. P a g e | 223 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry  Limiting stress condition 4. List out the shape factors for the rectangular, triangular, circular and diamond section.  Rectangular section, S = 1.5  Triangular section, S = 2.346  Circular section, S = 1.697  Diamond section, S = 2 5. Mention the types of frames.  Symmetric frames  Un-symmetric frames 6. What are symmetric frames and how they analyzed? Symmetric frames are frames having the same support conditions, lengths and loading conditions on the columns and beams of the frame. Symmetric frames can be analyzed by,  Beam mechanism  Column mechanism 7. What are unsymmetrical frames and how they analyzed? Un–symmetric frames have different support conditions, lengths and loading conditions on its columns and beams. These frames can be analyzed by,  Beam mechanism  Column mechanism  Panel or sway mechanism
27. 27. P a g e | 224 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry  Combined mechanism 8. What is the effect of axial force on plastic moment when a section is subjected to axial force? Thus far the cross sections considered are only carrying moment. In the presence of axial force, clearly some material must be given over to carry the axial force and so is not available to carry moment, reducing the capacity of the section. Further, it should be apparent that the moment capacity of the section therefore depends on the amount of axial load being carried. Considering a compression load as positive, more of the section will be in compression and so the neutral axis will drop. 9. Draw a stress strain curve for a perfectly plastic material. 10.What is a mechanism? When an n-degree indeterminate structure develops n plastic hinges, it becomes determinate and the formation of an additional hinge will reduce the structure to a mechanism. Once a structure becomes a mechanism, it will collapse.
28. 28. P a g e | 225 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 11.What are the different types of mechanism?  Beam mechanism  Column mechanism  Panel or sway mechanism  Gable mechanism  Combined or composite mechanism 12.What are the methods of plastic analysis?  Static method (Lower Bound Theorem)  Kinematic method (Upper Bound Theorem) 13.State the lower bound theorem or static theorem of plastic collapse. Lower bound theory states that the collapse load is determined by assuming suitable moment distribution diagram. The moment distribution diagram is drawn in such a way that the conditions of equilibrium are satisfied. 14.State upper bound theorem of plasticity. Upper bound theory states that of all the assumed mechanisms the exact collapse mechanism is that which requires a minimum load. 15.Define shape factor. The shape factor (S) is defined as the ration of the plastic moment of a section to the yielded moment of the section. The shape factor is also the ratio of plastic modulus of the section to the elastic modulus of the section. 𝑆 = 𝑀 𝑃 𝑀 𝑦 = 𝑍 𝑃 𝑍
30. 30. P a g e | 227 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 20.What is plastic moment? When the moment is further increased, there will be a stage at which all fibres from top to bottom of the section will completely yield and the section would not be able to take any further moment. The resisting moment corresponding to this fully plastic stage is called the plastic moment Mp. 21.Define plastic modulus of a section. The plastic modulus of a section is the first moment of the area above and below the equal area axis. It is the resisting modulus of a fully plasticized section. 𝑍 𝑝 = 𝐴 2 ( 𝑦1 + 𝑦2 ) 22.List the possible locations of plastic hinges in a structure.  Plastic hinges occurs under the loads  Plastic hinges occurs at joints 23.Define moment redistribution. Moment redistribution refers to the behavior of statically indeterminate structures that are not completely elastic, but have some reserve plastic capacity. 24.How is the shape factor for a hollow circular section related to the shape factor of an ordinary circular section? The shape factor of the hollow circular section = a factor K * shape factor of ordinary circular section.
31. 31. P a g e | 228 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 𝑆ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 ℎ𝑜𝑙𝑙𝑜𝑤 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = 𝑠ℎ𝑎𝑝𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 × ( 1 − 𝑐3 ) ( 1 − 𝑐4 ) 25.Give the governing equation for bending. The governing equation for bending is given by, 𝑀 𝐼 = 𝜎 𝑦 Where, M = bending moment I = moment of inertia σ = stress y = CG distance 26.What is meant by plastic analysis of structure? The analysis of beams and structures made of such flexural members are called plastic analysis of structure. 27.What is the difference between plastic hinge and mechanical hinge? Plastic hinges modify the behavior of structures in the same way as mechanical hinges. The only difference is that plastic hinges permit rotation with a constant resisting moment equal to the plastic moment Mp. At mechanical hinges, the resisting moment is equal to zero. 28.List out the assumptions made for plastic analysis.  Plane transverse sections remain plane and normal to the longitudinal axis before and after bending
32. 32. P a g e | 229 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry  Effect of shear is neglected  The material is homogeneous and isotropic both in the elastic and plastic state  Modulus of elasticity has the same value both in tension and compression  There is no resultant axial force in the beam  The cross section of the beam is symmetrical about an axis through its centroid and parallel to the plane of bending UNIT – 5 1. State the principle of super position of forces? When a body is subjected to a number of external forces, the forces are split up, and their effects are considered on individual sections. The resulting deformation, of the body is equal to the algebraic sum of the deformations of the individual sections. Such a principle of finding the resultant deformation is called the principle of superposition. 2. Define statically determinate structure. If the conditions of equilibrium (i.e.) ΣH=0, ΣV=0 and ΣM=0 alone are sufficient to find either external reactions or internal forces in a structure, the structure is called a statically determinate structure. 3. Define statically indeterminate structure. If the conditions of equilibrium (i.e.) ΣH=0, ΣV=0 and ΣM=0 alone are not sufficient to find either external reactions or internal forces in a structure, the structure is called a statically indeterminate structure.
33. 33. P a g e | 230 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 4. Define compatibility in force method of analysis. Compatibility is defined as the continuity condition on the displacements of the structure after external loads are applied to the structure. 5. Differentiate the statically determinate structures and statically indeterminate structures. S. NO STATICALLY DETERMINATE STRUCTURES STATICALLY INDETERMINATE STRUCTURES 1. Conditions of equilibrium are sufficient to analyze the structure Conditions of equilibrium are insufficient to analyze the structure 2. Bending moment and shear force is independent of material and cross sectional area Bending moment and shear force is dependent of material and independent of cross sectional area 3. No stresses are caused due to temperature change and lack of fit Stresses are caused due to temperature change and lack of fit 4. Extra conditions like compatibility of displacements are not required to analyze the structure. Extra conditions like compatibility of displacements are required to analyze the structure along with the equilibrium equations.
34. 34. P a g e | 231 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 6. Write down the rotation matrix for 2D truss element. In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix 7. Write down the compatibility equation used in flexibility matrix method. 𝑅 × ∆ = {𝐹} 8. Define force transformation matrix. The connectivity matrix which relates the internal forces Q and the external forces R is known as the force transformation matrix. Writing it in a matrix form, {Q} = [b] {R} Where, Q = member force matrix / vector b = force transformation matrix R = external force / load matrix / vector 9. What is transformation matrix? If, A and B are the matrices of two linear transformations, then the effect of applying first A and then B to a vector x is given by: (This is called the associative property.) In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices.
35. 35. P a g e | 232 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 10.Write down the stiffness matrix for 2D beam element. The stiffness matrix for a 2 D beam element is given by, 11.Describe the uses of force method. What are the basic steps in the force method to find internal forces in statically indeterminate structure? With the advent of computers, matrix methods of solving structures have become very popular. The behavior of a structure can largely be defined by defining the force – displacement relationship in the form of a matrix. Steps:  Applying a force on the structure  Working out the internal forces and moments  Computing displacement (and rotations) at specific locations making use of the values in the above step. 12.What are the basic unknowns in stiffness matrix method? In the stiffness matrix method nodal displacements are treated as the basic unknowns for the solution of indeterminate structures. 13.Define stiffness coefficient kij. Stiffness coefficient ‘kij’ is defined as the force developed at joint ‘i’ due to unit displacement at joint ‘j’ while all other joints are fixed.
36. 36. P a g e | 233 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 14.What is the basic aim of the stiffness method? The aim of the stiffness method is to evaluate the values of generalized coordinates ‘r’ knowing the structure stiffness matrix ‘k’ and nodal loads ‘R’ through the structure equilibrium equation. {R} = [K] {r} 15.What is the displacement transformation matrix? The connectivity matrix which relates the internal displacement ‘q’ and the external displacement ‘r’ is known as the displacement transformation matrix ‘a’. {q} = [a] {r} 16.How are the basic equations of stiffness matrix obtained? The basic equations of stiffness matrix are obtained as:  Equilibrium forces  Compatibility of displacements  Force displacement relationships 17.What is the equilibrium conditions used in the stiffness method? The external loads and the internal member forces must be in equilibrium at the nodal points. 18.What is meant by generalized coordinates? For specifying a configuration of a system, a certain minimum no of independent coordinates are necessary. The least no of independent coordinates that are needed to specify the configuration is known as generalized coordinates.
37. 37. P a g e | 234 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 19.Write the element stiffness for a truss element. The element stiffness matrix for a truss element is given by, 20.Write about the force displacement relationship. The relationship of each element must satisfy the stress-strain relationship of the element material. 21.What is the compatibility condition used in the flexibility method? The deformed elements fit together at nodal points. 22.Write the element stiffness matrix for a beam element. The element stiffness matrix for a beam element is given by, 23.Compare flexibility method and stiffness method. FLEXIBILITY MATRIX METHOD STIFFNESS MATRIX METHOD The redundant forces are treated as basic unknowns. The joint displacements are treated as basic unknowns The number of equations involved is equal to the degree of static indeterminacy of the structure. The number of displacements involved is equal to the no of degrees of freedom of the structure
38. 38. P a g e | 235 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry The method is the generalization of consistent deformation method. The method is the generalization of the slope deflection method. Different procedures are used for determinate and indeterminate structures The same procedure is used for both determinate and indeterminate structures. 24.Is it possible to develop the flexibility matrix for an unstable structure? In order to develop the flexibility matrix for a structure, it has to be stable and determinate. 25.What is the relationship between flexibility and stiffness matrix? The element stiffness matrix ‘k’ is the inverse of the element flexibility matrix ‘f’ and is given by f = 1/k or k = 1/f. 26.What are the types of structures that can be solved using stiffness matrix method? Structures such as simply supported, fixed beams and portal frames can be solved using stiffness matrix method. 27.Give the formula for the size of the global stiffness matrix. The size of the Global Stiffness Matrix (GSM) = number of nodes * degrees of freedom per node. 28.List the properties of the stiffness matrix.  It is a square matrix and always it should be a square matrix.  It is a symmetric matrix [𝑘] = [𝑘] 𝑇
39. 39. P a g e | 236 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry  The sum of elements in any column must be equal to zero.  It is an unstable element therefore the determinant is equal to zero.  The order of stiffness is equal to the number of co – ordinates. 29.List the properties of flexibility matrix.  Flexibility matrix is a square matrix of order nd nd  Flexibility matrix is a symmetrical matrix  Elements of flexibility matrix may be positive or negative except leading diagonal element which is always positive  Elements of flexibility matrix are displacements and they can be computed only if the structure is stable. If structure is unstable internally or externally, then displacements are indefinitely large and flexibility matrix does not exist. 30.Why the stiffness matrix method is also called equilibrium method or displacement method? Stiffness method is based on the superposition of displacements and hence is also known as the displacement method. And since it leads to the equilibrium equations the method is also known as equilibrium method. 31.Define a primary structure. A structure formed by the removing the excess or redundant restraints from an indeterminate structure making it statically determinate is called primary structure. This is required for solving indeterminate structures by flexibility matrix method.
40. 40. P a g e | 237 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 32.If the flexibility matrix is given as[𝑭] = [ 𝟐 −𝟏 −𝟏 𝟒 ]. Write the corresponding stiffness matrix. 𝑆𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠 𝑚𝑎𝑡𝑟𝑖𝑥 = 1 𝐹𝑙𝑒𝑥𝑖𝑏𝑖𝑙𝑖𝑡𝑦 𝑚𝑎𝑡𝑟𝑖𝑥 (i.e.) [𝐾] = [𝐹]−1 33.Define degree of kinematic indeterminacy (or) Degree Of Freedom. It is defined as the least no of independent displacements required to define the deformed shape of a structure. There are two types of DOF  Joint type DOF  Nodal type DOF 34.Briefly explain the two types of DOF.  Joint type DOF This includes the DOF at the point where moment of inertia changes, hinge and roller support, and junction of two or more members.  Nodal type DOF This includes the DOF at the point of application of concentrated load or moment, at a section where moment of inertia changes, hinge support, roller support and junction of two or more members.
41. 41. P a g e | 238 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 35.Name any two force methods to analyze the statically indeterminate structures.  Column analogy method  Flexibility matrix method  Method of consistent deformation  Theorem of least work 36.What are the different methods used to analyze indeterminate structures?  Finite element method  Flexibility matrix method  Stiffness matrix method 37.Write the formulae for degree of indeterminancy.  Two dimensional in jointed truss (2D truss) 𝑖 = (𝑚 + 𝑟) − 2𝑗  Two dimensional rigid frames/plane rigid frames (2D frame) 𝑖 = (3𝑚 + 𝑟) − 3𝑗  Three dimensional space truss (3D truss) 𝑖 = (𝑚 + 𝑟) − 3𝑗  Three dimensional space frame (3D frame) 𝑖 = (6𝑚 + 𝑟) − 6𝑗 Where, m = number of members r = number of reactions
42. 42. P a g e | 239 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry j = number of joints 38.Write the element flexibility matrix for a truss member. The element flexibility matrix (f) for a truss member is given by, 39.Briefly mention the two types of matrix methods of analysis of indeterminate structures.  Flexibility matrix method It is defined as the deformation produced for unit load. It is denoted by the symbols[𝑎] 𝑜𝑟 [𝑓] 𝑜𝑟 [𝛼]. This method is also called the force method in which the forces in the structure are treated as unknowns. The no of equations involved is equal to the degree of static indeterminacy of the structure.  Stiffness matrix method It is defined as the force required for unit deformations. It is denoted by the symbol[𝑘]. This is also called the displacement method in which the displacements that occur in the structure are treated as unknowns. The no of displacements involved is equal to the no of degrees of freedom of the structure.
43. 43. P a g e | 240 Prepared by R.Vijayakumar, B.Tech (CIVIL), CCET, Puducherry 40.Define flexibility influence coefficient. Flexibility influence coefficient (fij) is defined as the displacement at joint ‘i’ due to a unit load at joint ‘j’, while all other joints are not load. 41.Define element co – ordinates. Each element having a displacement along two directions (x and y) is said to be a element coordinates. 42.Define global co – ordinates. For the whole structure having a displacement along the two directions (x and y) is said to be a global coordinates. 43.Write the element flexibility matrix for a beam element. The element flexibility matrix (f) for a beam element is given by,