Influence lines for statically indeterminate structures. I. Basic concepts about the application of method of forces.

Transcription

1 Influence lines for statically indeterinate structures I. Basic concepts about the application of ethod of forces. The plane frae structure given in Fig. is statically indeterinate or redundant with degree of statical indeterinacy n=2, this structure has two redundant ebers. The internal force at any section of the presented plane frae could be obtained by the following general equation, using the ethod of forces: " S " = " S " S " X " S " X ",,,2 2 where S is bending oent, shear or noral force at section (or support reaction if the proble is to find its influence line); S is required internal force at section for the corresponding siple statically deterinate syste; S, is the internal force at section for the priary syste caused by the force (or oent) X, replacing the first eliinated constraint; S,2 is the internal force at section caused by the force (or oent) X 2 introduced to replace the second eliinated constraint. F= F= X X =const 4 Figure Statically indeterinate plane frae structure II. Deterination of influence lines Let us construct the influence lines for the internal forces in section of the plane frae structure given in Fig.. The bending oent, shear and noral forces at section can be expressed as: " M " = " M " M " X " M " X ",,,2 2 =,,2 2 =,,2 2 " Q " " Q " Q " X " Q " X ", " N " " N " N " X " N " X ". The following reasoning holds for the bending oent only, but they also could be done for shear and noral forces as well as for any support reaction. M, and M,2 are constants, which are dependant on the chosen siple syste, and respectively on the introduced unknown forces X and X 2, but they do not depend on the external loads, in that respect on the unit load oving across the road lane. M depends on the position of oving load, and corresponding 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 54

2 influence line for the priary syste should be derived. The forces X and X 2, which are introduced to replace the eliinated constraints depend on the external loads, therefore they are dependant on the position of the unit load. In the ethod of forces the basic unknowns are obtained by the canonical equations having the following appearance: X X Δ = 2 2 f X X Δ = f The coefficients ij to the unknowns of the canonical equations represent the generalized displaceents of the priary structure obtained by the eliination of the redundant ebers. These displaceents being due to unit loads (forces or oents) acting along the direction of eliinated constraints. Nuerically the values of these coefficients depend on the layout of the chosen priary structure and the cross sectional diensions of its ebers. The coefficient ik represents the displaceent along the direction i, induced by a unit action acting along the direction k. In line with the above stateents the coefficients ij do not depend on the external loads, in that respect for influence line construction, they do not depend on the position of load unity. The coefficients Δ if indicate the displaceents along the direction of eliinated constraints caused by the applied loads. As far as the applied loads are presented by the unit force oving along the road lane, for influence lines deterination, these displaceents,, depend on the position of the load unity. In that respect in order to obtain the unknown forces first construct the influence lines for displaceents Δ if. The canonical equations written in atrix for take the following appearance: X = Δ, [ ] { } { f } Wherefro follows that: X = Δ. { } [ ] { f } [ ] is so called copliance atrix of size 2x2 containing the coefficients ij of the free ters Δ if, { } X is a vector which consists of unknown forces X i. Let us introduce a atrix [ β ] in such a way that: [ β ] = [ ]. [ β ] is the inverse atrix of [ ] can be expressed as: X = β Δ, { } [ ] { f } or in an expanded for: " X" = β " Δ f " β2 " Δ2f ", " X 2" = β2 " Δ f " β22 " Δ2f ". The influence lines for the unknown forces Δ if, { f } X i we should Δ is the vector, ultiplied by. Then the unknowns of the ethod of forces X i could be derived fro the above expressions. 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 55

3 The chosen priary statically deterinate structure, obtained by eliinating all the redundant constraints, of the given one, is presented in Fig.. The forces X and X 2 introduced to replace these constraints are depicted in the sae figure. Next we should apply successively to the priary structure the unit actions X = and X 2 = and trace the diagras of the corresponding bending oents M i. These diagras are presented in Fig X =.75 M, =.5 Q, = M, =.5 N, = Q, =.25 N, = M M 2.25 X 2 =.25 Figure 2 Bending oent diagras to the siple structure What follows is the calculation of coefficients ij, ultiplying one by another the unit graphs M and M 2, in order to copose of copliance atrix = =, = =, = =. 3 The copliance atrix [ ] ultiplied by reads: 2 2 [ ] = After inversing the atrix [ ] and ultiplying the inverse atrix with we get the atrix [ ] [ β ] = Next we should construct the influence lines for the displaceents Δ if β :, as elastic curve of the road lane. For that purpose a unit load of the direction of required displaceent is introduced. The unit load coincides with actions X and X 2, the relevant bending oent diagras are M and M 2. 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 56

4 The conjugate beas with the corresponding fictitious loads are given in Fig. 3. The bending oent diagras in the conjugate beas are the required displaceents influence lines " Δ f " and " Δ 2 " (Fig. 3). f 3 3 X 2 = X = M M 2 Conjugate bea 3/() φ φ 2 φ φ 2 w w /() Staticaly deterinate conjugate bea 3/() /() f 2f Figure 3 Influence lines for the displaceents Now, the influence lines for unknown forces X i should be derived as { X} [ ] { f } lines are presented in Fig Δ if X X 2.25 = Δ. These Figure 4 Influence lines for the unknown forces Finally, the influence lines for internal forces at section of siple syste should be drawn. These graphics are depicted in Fig. 5. X i 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 57

5 " M " Figure 5 Influence lines for the internal forces in the priary statically deterinate syste The final influence lines, for required internal forces M, Q and N at section of the statically indeterinate structure, are obtained by the following equations: " M " = " M ".5 " X ".5 " X ", 2 " Q " = " Q ".75 " X ".25 " X ",.25 2 " N" = " N " " X" " X2". These lines are given in Fig " Q " " N " " M " " Q " " N " Figure 6 Influence lines for the internal forces in the original plane frae 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 58

6 III. Kineatical ethod of influence lines construction Kineatical way of influence line construction was exained in details for the trusses. Recall the MullerBreslau principle here which states that: The influence line for a function (M, Q, N or support reaction) coincides with the graphics of vertical displaceents of the road lane, obtained by application of unit virtual displaceent at the point (or points) of application of the function. For statically deterinate structure the deflected shape (respectively influence lines) are series of straight line segents. For statically indeterinate structures the influence lines are copound of several curves, one for each plate of the road lane. In order to obtain the influence line for bending oent at section, using the kineatic ethod, the plane frae should be first cut at this section by placing a hinge (the constraint of the bending oent is eliinated). Next a unit utual rotation is iposed in line of action of M in such a way that the oent perfors negative work (Fig. 7). If the original frae has n redundant constraints, the new structure, with one eliinated constraint, has n redundant ebers. Thus, we can obtain only the shape of influence line, or this is only qualitative representation of the influence line. η 2 M φ φ 2 η φ 2 φ M Δϕ=ϕ ϕ 2 = Figure 7 Kineatic ethod for bending oent influence line In order to obtain the value of influence line ordinates nuerically two different approaches are applicable. According to the first approach for calculation of the influence line ordinate η a unit load should be placed in the original frae structure and the bending oent in section for this frae should be coputed. The second approach is based on the ethod of forces. We can consider the odified structure with eliinated constraint as a statically indeterinate siple syste (siilar to statically deterinate siple syste). The bending oent M is the basic unknown of the force ethod in this case, with other words: X =M (Fig. 8), is utual rotation caused by M = (or X =). The applied load is load unity on the road line above the required ordinate, in that respect Δ f is a utual rotation of the direction of M induced by the unit load and can be derived by the following expressions: M M f Δ f =Σ ds, where Mand M f are bending oent diagras in (n) statically indeterinate syste (the syste with eliinated constraint). Alternatively the sae utual rotation can be obtained as: 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 59

7 M M f ds Δ f =Σ, here M f is bending oent diagra in any statically deterinate syste, obtained fro the corresponding (n) statically indeterinate syste (Fig. 8). Therefore for convenience the unit force can be applied in any statically deterinate siple syste obtained by eliinating the redundant constraints of the odified syste. The canonical equation reads: X Δ f = respectively M = Δ f or finally M = Δ f /. The required ordinate η becoes: η = Δ f / M = 2. F= F=.. M.4286 M f, M f,2. Figure 8 Deterination of influence line ordinates M M =Σ ds = /( ), M M f, Δ f =Σ ds = 2.575/( ), M M f,2 Δ 2 f =Σ ds =.5747 /( ), η = Δ f / =.5625, η2 = Δ 2 f / =.25. The ordinate values are absolutely the sae as those given in Fig. 4. In general if nuerical values of influence line are to be deterined, we can copute the displaceents at successive points along the road lane, when the structure is subjected to the unit load placed at the eliinated constraint (utual rotation caused by unit force is equal to the vertical displaceent induced by the bending oent M according to the Maxwell s theore). Then each obtained value of vertical displaceents, of the points belonging to the road lane, ust be divided by the displaceent at the point where the unit load acts, taken with negative sign (in our case this displaceent is denoted ). In conclusion it can be said that the kineatic ethod perits the easy deterination of the shape of the influence line for any action, this shape being the sae as that of the elastic curve of the corresponding structure with eliinated constraint. This analogy can be considerable value both in checking the accuracy of influence lines obtained by other ethods and in seeking those parts 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 6

8 of the structure which ust be loaded in order to provide the axiu or iniu values of the required internal forces. Influence lines for X and X 2 using the kineatic ethod. X = X X 2 φ X 2 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 6

9 References DARKOV, A. AND V. KUZNETSOV. Structural echanics. MIR publishers, Moscow, 969 WILLIAMS, А. Structural analysis in theory and practice. ButterworthHeineann is an iprint of Elsevier, 29 HIBBELER, R. C. Structural analysis. PrenticeHall, Inc., Singapore, 26 KARNOVSKY, I. A., OLGA LEBED. Advanced Methods of Structural Analysis. Springer ScienceBusiness Media, LLC 2 2 S. Parvanova, University of Architecture, Civil Engineering and Geodesy Sofia 62