Structural Analysis 2 Lecture 4: INFLUENCE LINES OF STATICALLY INDETERMINATE STRUCTURES BY THE FORCE METHOD Maxwell's theorem of reciprocal displacements The method is based on Maxwell's law of interchangeability which is often used in mechanics: the interchangeability of internal force and external displacement. β¦ Betti’s law: by virtual force theorem β¦ π =π , , , +π =0 , , β¦ π , = π , , +π , , = 0 β¦ Internal complementary work by definition : π , , =π , , =∫ ππ + ∫ ππ + ∑ β thus π , , =π , , β¦ Maxwell’s law : the force-system P is only a single force and π = 1, and similarly Q is a single force and π=1 β¦ π β¦ π , , = ππ =π , , = ππ =π The type of movement corresponding to the tested internal force is inserted into the examined point and the vertical displacement diagram of the support is drawn. Maxwell's theorem of reciprocal displacements Betti’s law: Maxwell’s law: Influence line diagram of a displacement: π₯ π π π , , =π , , π=1 π’ π=1 π’ π’ =π’ π π π’ π=1 π=1 π π’ π π π’ = π , Influence line diagram of a displacement What are we looking for? What do we use? The influence line diagram of the vertical displacement at the cross-section K: π π A unit vertical force at the cross-section K: πΉ = 1 The influence line diagram of the horizontal displacement at the crosssection K: π π A unit horizontal force at the crosssection K: πΉ = 1 The influence line diagram of the rotation at the cross-section K: π π A unit positive moment at the crosssection K: π = 1 The influence line diagram of the relative displacement of points A and B: π π’ A par of unit forces on the lines of influence passing through the two points : πΉ = ±1 The influence line diagram of the relative rotation of joint C: π π A par of unit moments on both sides of the hinge: π = ±1 What do we count? How? Diagram of the vertical displacement (deflection) of the structure Small displacements, moment load method, virtual force theorem, or a combination of these Force method – influence lines Statically indeterminate structures Force method – influence lines As a result of the moving unit force, the load factors change continuously. So, we need to define the effect diagrams of the load factors in order to get the influence line diagrams of the connection forces. The system of geometric (compatibility) conditions/equations based on principles of superpositioning: β¦ ∑ π π π +π π = 0 , π = 1. . π. β¦ Here the unknowns are the influence line diagrams π π of the connection/redundant forces π . β¦ The π unit factors and the coefficient matrix formed from them are the same as for a fixed/unmoving load. β¦ The effect diagrams of the load factors π π represent the influence line diagrams of the displacements occurring at the location of the redundant forces. Statically indeterminate structures Force method – influence lines The influence line diagrams of the redundant forces are given by the solution of the conditional equation system: β¦ π π = −∑ π§ π π , π = 1. . π. β¦ π§ means the impact factors. With the action diagrams , any reaction force, internal force or displacement can be determined at any cross section of the structure: β¦ π πΆ =π πΆ +∑ πΆ π π . β¦ π πΆ is the influence line of a mechanical property of a cross-section K of the base structure, β¦ πΆ the volume of a mechanical property from the applied unit redundant force at the position/case of j. β¦ πΆ may be replaced by any mechanical feature : π , π , π , π , π , π΅, … Force method – influence lines Steps 1st step: primary/base structure determination. The considerations set out for stationary loads are applied. 2nd step: preparation of the flexibility coefficient matrix. (Just like with a stationary load.) 3rd step: preparation of influence line diagrams of load factors πΌ πππ . β¦ The load factors are the displacements at the location of the cuts. β¦ Based on Maxwell's interchangeability theorem, their effect diagrams are identical to the vertical displacement diagram created by a (virtual) unit force work-compatible with the displacement : π π = π π . (π π = π figure from π = 1 .) 4th step: determination of the influence line diagrams of the redundant forces πΌ πΏπ . β¦ This step requires determining the inverse of the coefficient matrix. β¦ The elements of this are the so-called π§ factors. The resulting π π effect diagrams are curved line diagrams. 5th step: the preparation of an influence line diagram determining a mechanical feature πΌ πͺπ² . β¦ We perform the principle of superposition in calculation for each ordinate. (Just like in case of a standing load.) β¦ The resulting figures are curved figures, that also show the characteristics of the figures of statically determinate structures: unit jump in the shear force diagram, v-shaped fracture in the moment diagram, locations of zero points, etc.. Illustrative examples E1: structure indeterminate to the first degree π πΆ π π π =− π π π =π πΆ + πΆ π π E2: structure indeterminate to the 2DOF Inverse of a 2x2 matrix: π π π = π π −π −π π E2: structure indeterminate to the 2DOF π πΆ =π πΆ + πΆ π π Example E3: branched holder indeterminate to the first degree Determine the moment influence line diagram of the cross-sections 3 and 6 of the beam, and , and the reaction force influence line diagram at point B! kNm2, 1 , 2 π 3 4 m/kN. 5 6 7 πΈπΌ πΈπΌ A C πΈπΌ B 16 m 8m 6m E3: branched holder indeterminate to the first degree 1st step: Free body diagram, primary/base structure determination. 1 2 3 5 6 7 πΈπΌ πΈπΌ A: 4 π =1 πΆ=2 πΈπΌ π΅=3 16 m 8m 6m E3: branched holder indeterminate to the first degree 2nd step: preparation of a matrix containing the unit factors . m3 Moment diagram from unit force π : 1 A: 2 3 πΈπΌ π =1 4 5 6πΈπΌ πΆ=2 16 kNm πΈπΌ π΅=3 16 m 7 8m 6m E3: branched holder indeterminate to the first degree 3rd step: preparation of influence lines of load factors ππ . Based on Maxwell's interchangeability theorem, this is a diagram of the vertical displacements caused by a unit load.. 7 5 π 6 1 2 3 π 4 The moment figure from the unit load π : π = π = = 64 m2 = 128 m2 4 8 12 π = π = 42,67 m2 8 16 kNm B π = π = 21,33 m2 1 The vertical displacement figure from the unit load π = π π : C 2 3 4 5 6 π =π B 7 E3: branched holder indeterminate to the first degree 4th step: creating the influence lines of the contact forces πΌ πΏπ . Calculation of πππ = πΌ πππ i πΌ πππ πΌ πΏπ = − πΌ πππ πππ 1 −42,67 16 − 128 16 2/3 -2048 0,911 2 −42,67 12 − 4 12 6 − 12 /2 8 -1376 0,612 3 −42,67 8 − 8 8 4 − 8 /2 8 2/3 -768 0,342 4 −42,67 4 − 12 4 2 − 4 /2 4 2/3 -288 0,128 5 0 0 0 6 +21,33 4 − 8 4/2 4/3 64 -0,028 7 0 0 0 1 2 3 4 5 6 7 5 6 7 πΌ πππ 1 2 3 4 πΌ πΏπ E3: branched holder indeterminate to the first degree 5th step: preparation of an influence line diagram determining a mechanical feature . β¦ So three concerning influence line diagram must be created: π π , π π and π π΅ . β¦ The tree multipliers/weighting values per unit load must be determined : π , π and π΅ . β¦ Then the influence line diagrams must be prepared according to the formula. π² . E3: branched holder indeterminate to the first degree π πΆ 1 2 3 4 5 6 1 -4 3 2 2 5 6 7 π π 3 1 4 4 5 π π 6 + πΆ π π 7 π π -8 =π πΆ 7 π π πΌ πΏπ π -8 0,911 -4 0,612 -0,896 0 0,342 2,736 0 0,128 1,024 0 0 0 0 -0,028 -0,224 0 0 0 +8 π π -0,712 E3: branched holder indeterminate to the first degree π πΆ 1 2 3 4 5 6 1 -6 2 -4 3 -2 5 4 π π 5 1 2 3 4 π π 6 7 2 6 + πΆ π π 7 π π -8 =π πΆ 7 π ππ πΌ πΏπ ππ π π -8 0,911 +8 -6 0,612 -1,404 -4 0,342 -1,264 -2 0,128 -0,976 0 0 0 2 -0,028 1,776 0 0 0 -0,712 E3: branched holder indeterminate to the first degree π πΆ 1 2 3 4 5 6 3 1 2 2,5 2 π π© 3 4 5 6 2 1,5 1 0,5 3 4 7 π π΄π 5 π π΄π 6 + πΆ π π 7 π π 1 =π πΆ 7 πΌ πΏπ π -3 π π΅ 3 0,911 0,267 2,5 0,612 0,664 2 0,342 0,974 1,5 0,128 1,116 1 0 1 0,5 -0,028 0,584 0 0 0 Literature In Hungarian: β¦ Kurutzné Kovács Márta: Tartók statikája, MΕ±egyetemi kiadó 2006. β¦ Pásztor Erzsébet, Tamássy Tamás: Tartók statikája példatár I., Tankönyvkiadó Budapest 1992. J9-1275. (source of illustrations on a yellow background) offered! β¦ BME, PTE MIK órai jegyzetek β¦ Gáspár Zsolt, Tarnai Tibor: Statika, egyetemi jegyzet, MΕ±egyetemi Kiadó, Budapest 2006. In English: R.C. Hibbeler: Structural Analysis, 9th edition, chapter 10 offered!