Torsion and shear stresses due to shear centre eccentricity in SCIA Engineer Delft University of Technology. Marijn Drillenburg

Transcription

1 Torsion and shear stresses due to shear centre eccentricity in SCIA Engineer Deft University of Technoogy Marijn Drienburg October 2017

2 Contents 1 Introduction Hand Cacuation Resuts in SCIA Investigation Pan 6 3 Geometry I-shaped cross section L-shaped cross section Findings Boundary Conditions Cantiever Beam Doube camped beam Findings Loading Pure bending Point oad in norma force centre Point oad in shear centre Findings Concusion Geometry Boundary Conditions Loading Recommendations 26 Appendices 28 A Torsion Theory 29 B Framework Anaysis 34 B.1 Overview B.2 Generation of input fie B.3 The Kerne B.4 GUI C Mape Fies 42 C.1 Cross sectiona properties C.2 L-shaped cross section oaded in pure bending C.3 L-shaped cross section oaded in norma force centre

3 Chapter 1 Introduction There is a suspicion that the framework anaysis program SCIA Engineer is not computing torsiona stresses and deformations in the correct way. To iustrate how this suspicion came to ight, this chapter provides the reader with an exampe structure in which torsion occurs. The structure wi be anaysed theoreticay first, by doing a hand cacuation using the theory described in Appendix A. This cacuation wi produce expected stress vaues. These expectations wi be compared to resuts from an anaysis of the structure in SCIA Engineer. Based on these resuts, a pan wi be presented to further anayse the possibe probems. 1.1 Hand Cacuation We wi first focus on a doube camped beam, oaded with point oad in the norma force centre at a distance 1 on the span. The cross section is L-shaped and pictured in Figure 1.2. Since this cross section is not symmetrica, we know from Appendix A this type of oading wi not ony cause bending moments, but aso a torsiona moment. This is caused by shear centre eccentricity in the non-symmetrica cross section. A schematic drawing of the structure is dispayed in Figure 1.1. Figure 1.1: Doube camped beam with torsiona moment Figure 1.2: L-shaped Cross Section Tabe 1.1: Materia Properties Symbo S235 Young s moduus E N/mm 2 Shear moduus G N/mm 2 2

4 The ocation of the norma force centre and the shear force centre have aready been computed. This cacuation was done by hand at first. Then the cacuation was done again in Mape. The cacuation in Mape yieded the same resuts. To save time, from now on Mape is used for the cacuations. A Mape fie where the properties for this cross section are cacuated is attached under the section L-shaped in Appendix C.1. For this exampe, we are interested in the interna torsiona moments in the two parts of the beam. We name the torsiona moment caused by eccentricity of the point oad T. If we sove this structure in terms of T, 1, 2 and the cross sectiona properties, we can ater vary the different parameters easiy and consider different positions for the oad. Because this is a static indeterminate structure, we can not simpy sove this probem using equiibrium. We need an additiona equation. In this exampe we use the fact that the rotation ϕ x1 just eft of the appied moment has to be equa to the rotation ϕ x2 just to the right. First we divide the beam into two parts, as shown in Figure 1.3. Figure 1.3: Doube camped beam divided in two parts The assumed moments M x1 and M x2, and the externay appied torque T have to satisfy the equiibrium equation where the tota moment around the node is 0. This eads to the first equation: T + M x1 M x2 = 0 (1.1) We aready know the constitutive reation between externay appied torque and rotation of a cross section. The second equation becomes: Soving for the unknown moments yieds ϕ x1 = ϕ x2 M x1 1 = M x2 2 (1.2) GI t GI t 2 M x1 = T M x2 = T (1.3) This eads to the moment distribution as shown in Figure 1.4. From these resuts it can be concuded that in a situation where a beam is oaded in torsion somewhere aong the span, 3

5 Figure 1.4: Moment distribution of doube camped beam the distribution of the torsiona moment can be determined by the equations from (1.3). The shear stresses that wi occur are inear in M t, so they wi aso foow this distribution. We wi consider three oad cases: Load at midspan Load at 1/4 span Load at 1/10 span For each of these oad cases, we wi determine the expected shear stresses and compare them to the resuts in SCIA. The computation of the cross sectiona properties and torsiona moment can be found in the Mape-fie in Appendix C.1. We reca the theory for thin waed open eements from Appendix A, and fi in equation (A.11) to find the maximum shear stress in the cross section: τ max = M t s c 1 2 I t = Nmm t mm 4 = 5.98N/mm 2 (1.4) Now we can scae this maximum vaue to the different oad scenarios, using the formuas from equation (1.3). The expected shear stresses are now known, and are reported in tabe 1.2. Note that τ 1 represents the maximum shear stress eft of where the oad is appied and τ 2 represents the maximum shear stress on the right. Tabe 1.2: Expected shear stresses τ 1 [%] τ 1 [N/mm 2 ] τ 2 [%] τ 2 [N/mm 2 ] Load at midspan 50% % Load at 1/4 span 75% % Load at 1/10 span 90% % Resuts in SCIA The structure is now imported in SCIA using the dimensions and materia properties defined above. The shear stresses are evauated for every oad scenario presented in the former section. In SCIA we have to choose in which fibre the shear stress has to be dispayed. Fibre 14 is chosen. This fibre is shown as a bue dot in Figure 1.2. The vaues for the shear stress are reported in the bod coumn abeed τ xy /τ xs. 4

6 Figure 1.5: Stress resuts with oad at midspan Figure 1.6: Stress resuts with oad at 1/4 span Figure 1.7: Stress resuts with oad at 1/10 span A comparison between the theory and the resuts from SCIA is presented in tabe 1.3. It can be observed that none of the vaues from SCIA correspond to their expected vaue. In most cases the vaue in SCIA is higher. In two of the cases, the vaue reported in SCIA is ower than the expected vaue. For exampe, the τ 2 for the oad at 1/10 span is 60% ower than the expectation. Tabe 1.3: Comparison between theory and SCIA Load at midspan Load at 1/4 span Load at 1/10 span τ 1,exp τ 1,scia Diff τ 2,exp τ 2,scia Diff [N/mm 2 ] [N/mm 2 ] [N/mm 2 ] [N/mm 2 ] % % % % % % When considering engineering purposes, this discrepancy can ead to dangerous situations. If the shear stresses occurring in a structure are 60% higher than their expected vaue, safety may not be guaranteed. Therefore it shoud be investigated why the vaues in SCIA do not correspond with the theory. 5

7 Chapter 2 Investigation Pan To investigate why SCIA is not dispaying the vaues as we woud expect, the foowing pan is presented to eiminate potentia issues. The first factor we wi consider is the geometry of the cross section. Without considering oading or boundary conditions, the cross sectiona properties for various shapes wi be anaysed. Important steps in this process are the determination of the norma force centre and the shear force centre. The cacuation wi be done by hand at first, whie comparing it to the vaues observed in SCIA ater. This check wi eiminate issues with the geometry, which can cause wrong resuts of the torsion cacuation. The second variabe we wi check for issues are the boundary conditions. For a simpe oading scenario, we wi anayse different boundaries. Again we wi first cacuate the vaues by hand. Then a cacuation in SCIA is performed, and the two resuts wi be compared. This check wi determine whether the torsiona deformation and stress resuts for a basic exampe are in ine with the theory. Another goa of this section is to check whether the distribution of the interna torsiona moment is performed correcty in SCIA. The fina parameter we wi check is the oading. In this part of the investigation we wi consider different oading situations. Whie we keep the cross section and boundary conditions fixed, we wi vary the oading type and check for differences between a hand cacuation and the resuts from SCIA. This check wi provide insight in whether the type of oading infuences the torsion resuts. By foowing the steps described in this investigation strategy, answers can be formuated to the foowing questions: Does SCIA cacuate cross sectiona properties in the correct way? Do different boundary conditions infuence the torsion resuts in SCIA? Is the discrepancy between the theory and SCIA infuenced by varying the oading? These answers wi provide the reader with a better understanding of the source of the probem with the torsion cacuation. After the different aspects of this probem are investigated, the ogica next step is to start thinking about soutions to the potentia issue. To begin this thought process, it is important to have a better understanding of how the software package operates. Therefore research is performed on frame anaysis software in genera. This wi be done by means of considering a very simpe exampe of a framework structure, for which a program is written to sove the interna force distribution and the defections. After this procedure, the issues in the program can possiby be expained. This eads to the fina question: Can an advice be formuated to the deveopers of SCIA Engineer to reduce discrepancy in vaue between a theoretica approach and an anaysis in SCIA of a torsion cacuation? 6

8 Chapter 3 Geometry Determining the characteristics of the cross section is one of the first steps SCIA takes in the evauation process. This procedure is performed in the Graphica User Interface, as described in Appendix B. An issue here coud cause wrong resuts in a stress cacuation, so it makes sense to investigate whether the vaues produced in SCIA are correct. To eiminate an issue with the cacuation of the cross sectiona properties we wi cacuate various cross sections by hand first, and compare them to the vaues obtained from SCIA. The manua provided by SCIA(Section on determination of standardised cross sectiona properties) provides us with the soution strategy. The cross sectiona properties are determined in two parts. The vaues that are cacuated in these two parts are: Part I: Biaxia bending and axia force Area Centre of gravity Ange of the principa axis system Principe moments of inertia Part II: Torsion Shear Centre Torsiona moment of inertia Warping constant Warping ordinate A good strategy to check whether the Part I properties are correct, is comparing the moments of inertia I yy, I zz and I yz for each cross section. Since determining the warping constant and the warping ordinate both are outside the scope of this paper, we wi focus on finding the position of the shear centre and the magnitude of the torsiona moment of inertia when investigating the Part II information. We wi consider two different cross section types: Symmetrica in two axes Symmetrica in one axis As mentioned before, for each type a comparison between the theory and the vaues from SCIA wi be made. If the vaues correspond to their expected vaue, we can eiminate an issue here. If the vaues do not correspond, further investigation is necessary. 3.1 I-shaped cross section The first cross section type we wi consider is a doube symmetrica I-shaped section. The shape and dimensions are presented in Figure 3.1. Since this cross section is symmetrica in two axes, we know the shear centre shoud coincide with the norma force centre. We aso expect to see a zero magnitude of I yz, due to symmetry. The cacuation of the cross sectiona properties is 7

9 Figure 3.1: I-shaped cross section performed in Mape, for checking this cacuation a reference is made to Appendix C.1. The resuts from SCIA can be observed in Figure 3.2. A comparison between these resuts is made in tabe 3.1. As can be observed from this tabe, the vaues in SCIA show no discrepancies with the resuts from the theory. Figure 3.2: Cross sectiona properties of I-shaped in SCIA 3.2 L-shaped cross section Now we wi consider a cross section that has ony one axis of symmetry. The ocation of the shear centre wi therefore not coincide with the norma force centre. The cross section we wi discuss is L-shaped, and has dimensions as pictured in Figure 3.3. Again the cross section is first 8

10 Tabe 3.1: Comparison theory and SCIA for I-shaped cross section Expectation SCIA Centroid (ync,znc) (100mm,125mm) (100mm,125mm) Area A 6300 mm mm 2 Moment of inertia y-axis I yy mm mm 4 Moment of inertia z-axis I zz mm mm 4 Moment of inertia combined I yz 0 0 Torsiona moment of inertia I t mm mm 4 Eccentricity Shear Centre (dy,dz) (0,0) (0,0) Figure 3.3: L-shaped cross section anaysed by hand. This hand cacuation is reported in the form of a Mape-fie in Appendix C.1. The ocation of the shear centre is determined using the theory that for thin-waed L- shaped cross sections, the shear centre ies on the connection between the two fanges(par. 5.5, Toegepaste Mechanica dee 2, Hartsuijker 2003)[1]. Because the eccentricity of the shear centre with respect to the norma force centre is given in the principa coordinate system in SCIA, we have to make an additiona cacuation. From Introduction into Continuum Mechanics [2] we know the transformation rue for rotating coordinate axes: [ ] ynew = z new [ cos(α) ] sin(α) sin(α) cos(α) [ ] yod z od Using these facts the theoretica resuts are obtained. The cross section is aso generated in SCIA, the resuts from that cacuation can be seen in Figure 3.4. Now the vaues can be compared in Tabe 3.2. Tabe 3.2: Comparison between resuts from the theory and SCIA for an L-shaped cross section Expectation SCIA Centroid (ync,znc) (71.71mm,71.71mm) (71.71mm,71.71mm) Area A mm mm 2 Moment of inertia y-axis I yy mm mm 4 Moment of inertia z-axis I zz mm mm 4 Moment of inertia combined I yz mm mm 4 Torsiona moment of inertia I t mm mm 4 Eccentricity Shear Centre (dy,dz) (83.74mm,0) (83.97mm,0) From Tabe 3.2 it can be concuded that a Part I information is accurate. However, the shear centre eccentricity is not the same as we have cacuated by hand and shows a sma discrepancy. The difference in vaue is 0.23 mm. (3.1) 9

11 Figure 3.4: Cross sectiona properties of L-shaped in SCIA 3.3 Findings In this chapter we have anaysed two types of cross sections: Symmetrica in two axes Symmetrica in one axis The vaues obtained by making a hand cacuation have been compared to the resut from a cacuation in SCIA. In section 3.1 we have estabished that for cross sections that are symmetrica in two axes, the resuts in SCIA match the theoretica expectation. In section 3.2, an anaysis was made on an L-shaped cross section. This type of cross section is ony symmetrica in one axis, which means the shear centre does not coincide with the norma force centre. We observed that the vaues of most cross sectiona properties matched the resuts in SCIA. However, the shear centre eccentricity reported in SCIA differed from the expectation by 0.23 mm. Further investigation is necessary to find out where this discrepancy originates. 10

12 Chapter 4 Boundary Conditions This chapter investigates the infuence of different boundary conditions on the resuts for torsion anaysis. In order to do this, a simpe oading scenario is assumed. The L-shaped cross section from Section 3.2 is used. Since we are interested in reviewing the torsiona deformation and shear stresses caused by torsion, a torsiona moment M t around the x-axis is appied. A beam eement with constant ength is then anaysed for two different boundary conditions. The first boundary condition that wi be considered, is a simpe fixed boundary. This type of structure is aso known as a cantiever beam. This exercise wi estabish whether SCIA cacuates shear deformation and shear stresses in a basic torsion probem as we woud expect. After this check has been carried out, we wi focus on an exampe where the same beam is camped on both ends. This wi ead to a doube camped structure. With that structure, we wi be abe to determine whether the torsiona moment is divided correcty over the beam, and aso whether the shear stresses have the same magnitude as we woud expect. 4.1 Cantiever Beam The first exampe that wi be discussed is a singe camped cantiever beam. The cross section is L-shaped, and the structure is oaded with a moment Mt = 5.0kN around the x-axis. A schematic overview of the structure is presented in Figure 5.1. Figure 4.1: Cantiever beam with L-shaped cross-section Since the cross sectiona properties have aready been determined in Section 3.2, we can now cacuate the maximum shear stress τ max and torsiona deformation ϕ x using the formuas from 11

13 Appendix A: τ max = Nmm 12.5mm mm 4 = N/mm Nmm 5000mm ϕ x = N/mm mm = mrad 4 Now the expected vaues are known, we can focus on the resuts from SCIA. The geometry and cross section of the structure are imported, and a cacuation is performed. The maximum shear stress and torsiona deformation according to SCIA are dispayed in Figure 4.3 and Figure 4.4 respectivey. (4.1) Figure 4.2: 3D-view of deformed L-shaped cantiever oaded in pure torsion Figure 4.3: Stress resuts for L-shaped cantiever oaded in pure torsion Figure 4.4: Deformation resuts for L-shaped cantiever oaded in pure torsion Now we can compare the resuts. Looking at tabe 4.1, we observe that the vaues from SCIA match the expected vaues very we. Tabe 4.1: Comparison between theory and SCIA resuts for L-shaped cantiever Theory SCIA τ max N/mm N/mm 2 ϕ x mrad mrad 12

14 4.2 Doube camped beam We wi consider a beam that is camped on both ends, simiar to the exampe from the introduction. The cross section and the ength of the beam are the same as in the previous exampe. Again the structure wi be oaded with a torsiona moment ony, but we wi consider three different positions for the oad. The goa is to investigate whether the distribution of the torsiona moment, and subsequenty the shear stresses, are correct. The structure we wi anayse is pictured in Figure 4.5. Figure 4.5: Doube camped cantiever beam with L-shaped cross-section Expectation We choose the oading positions the same as in the introduction, so at 1/2, 1/4 and 1/10 of the span. These positions are pictured in Figure 4.6. Note that the dotted arrows indicate the position of the torsiona moment, not to be confused with the ocation of a point oad. The moment distribution due to these oading scenarios can be obtained by using the same soution strategy as in the introduction. There we found the torsiona moment distribution of a doube camped structure is directy proportiona to the position of the oad. Appying equation (1.3) to this scenario three separate moment distribution graphs can be drawn. These can be found in Figure 4.7. With this moment distribution the shear stresses can be obtained by fiing in equation (A.11). 13

15 Figure 4.6: Loading positions doube camped cantiever Figure 4.7: Expected moment distributions Resuts in SCIA The interna moment distribution given by SCIA matches our expectation. The resuts can be seen in Figure 4.8. The magnitude of the interna forces is correct, but the sign is different. Where we expected to see a positive vaue, SCIA reports a negative and vice-versa. This is due to the fact that SCIA can ony dispay the vaues in the oca coordinate system of the cross section, which is different than the coordinate system we have used in the cacuation. Figure 4.8: Interna moment distribution of doube camped beam in SCIA The other check that needs to be done is whether this torsiona moment eads to the correct shear stresses. Since the shear stresses are inear in M x, they can be easiy computed. The shear stress diagrams from SCIA can be found in Figure 4.9. A comparison between the expected shear stresses and the resuts from SCIA is made in tabe 4.2. Again the sign of the stresses is different than we woud expect, but this can be attributed to the coordinate system choice. The magnitudes of the shear stresses do match our expectation. Figure 4.9: Shear stress resuts for doube camped beam in SCIA 14

16 Tabe 4.2: Shear stress comparison for doube camped beam Expectation Shear stress in SCIA part 1 part 2 part 1 part 2 Load midspan N/mm N/mm N/mm N/mm 2 Load 1/4 span N/mm N/mm N/mm N/mm 2 Load 1/10 span N/mm N/mm N/mm N/mm Findings From the anaysis of a singe camped cantiever beam, we have observed that the vaues of a cacuation in SCIA match the expected vaues. Therefore we can say that the computation of torsiona deformation and shear stresses for a basic torsion probem are correct. From the anaysis of the doube camped beam in section 4.2, we have seen that the distribution of the interna torsiona moment is correct in SCIA. This was determined using the shear stress distribution due to a torsiona moment appied on different points on the span. 15

17 Chapter 5 Loading The next step in the investigation is to consider the oading. Because there are no significant issues when considering geometry and boundary conditions, from now on the focus wi be on different oading scenarios. Again we wi consider a singe camped beam, which can be considered as a simpe boundary condition. The same L-shaped cross section is chosen as in previous exampes. We have aready estabished in section 4.1 that for pure torsion, the resuts for this structure are in ine with the theory. To investigate further, this chapter wi discuss three different oad cases: Pure bending with externa torque Point oad in norma force centre Point oad in shear centre 5.1 Pure bending Figure 5.1: Cantiever beam with L-shaped cross-section The structure is oaded with a moment T y around the y-axis at the end of the span. Since this cross section is non-symmetrica, the moment of inertia I yz is non-zero[3]. This means that in this exampe doube bending wi occur. For carity reasons, ony the main steps of the cacuation wi be shown in this section. For the compete cacuation, reference is made to Appendix C.2 where a Mape fie can be found. The constitutive reation for bending[3] is: [ ] My = M z [ ] EIyy EI yz EI yz EI zz The cross-sectiona properties have aready been determined in section 3.2. These magnitudes can be found in tabe 3.2 and the Young s moduus for stee S235 in tabe 1.1. Aso knowing [ κy κ z ] (5.1) 16

18 the moment distribution we obtain the foowing system: [ ] [ = Nmm Nmm 2 Nmm 8.7 Nmm Nmm 2 Eaborating this we obtain the curvatures in y- and z-direction: [ ] [ ] κy = κ z ] [ ] κy κ z (5.2) (5.3) Now we can cacuate the defections using this curvature distribution. This is done using the theory from section 8.4 of Hartsuijker[1]. The resut is: [ ] [ ] uy mm = (5.4) u z mm The next step is to ook at the resuts in SCIA. These resuts are presented in Figure 5.2 and 5.3. As expected, doube bending occurs. The magnitudes of the defection at the end of the cantiever are exacty the same as in equation (5.6). The sign of both defections is opposite of what we have cacuated. This is due to the fact that SCIA dispays the defections in the coordinate system of the cross section, which is different than the coordinate system used in the hand cacuation. Figure 5.2: 3D-view of defected L-shaped cantiever oaded in pure bending in SCIA Figure 5.3: Tabe of defections of L-shaped cantiever oaded in pure bending in SCIA 17

19 5.2 Point oad in norma force centre Figure 5.4: Cantiever beam oaded with a point oad in the norma force centre Tabe 5.1: Materia Properties Symbo S235 Young s moduus E N/mm 2 Shear moduus G N/mm 2 We wi consider a cantiever with a L-shaped cross section, as shown in figure 5.4. The materia used for this structure is described in tabe 5.1. The force F acts in the norma force centre. We know from the theory(section 5.5 of Hartsuijker[1]) that in L-shaped cross sections the norma force centre does not coincide with the shear force centre. Therefore the expectation is that in this case the force F wi not ony cause a oad in Z-direction, but wi aso ead to a torsiona moment. This torsiona moment wi cause shear stresses and a deformation around the x-axis. The cacuation of the expected deformations is performed in two parts. First the defection due to bending ony is determined. Then the torsiona deformation is cacuated. Aso the expected maximum shear stress and interna moment are cacuated. The compete cacuation procedure is presented in Appendix C.3, but for carity reasons ony the resuts are presented here. The expected deformations in y-,z- and x-direction are: u y mm u z = mm (5.5) ϕ x mrad The expected interna forces due to shear centre eccentricity are: [ ] [ ] Mx = 6 Nmm 5.98 N/mm 2 τ max The resuts from SCIA are presented in Figures 5.5, 5.6, 5.7 and 5.8. The vaue for maximum shear stress was found with the option consider torsion due to shear centre eccentricity enabed. Turning this option off, the vaues in coumn τ xy /τ xs a reduced to zero. Now we coect the expectations and the resuts from SCIA, to compare the magnitudes in tabe 5.2. From the 3D-overview and the deformation resuts, we observe no rotation around the x- axis. The defection due to bending in y-direction is correct, but the vaue in z-direction shows some discrepancy. No torsiona moment is reported. And finay, the maximum shear stress that SCIA reports is aso different in vaue than our expectation. (5.6) 18

20 Figure 5.5: 3D-view of deformation of L-shaped Cantiever with oad in NC Figure 5.6: Deformation resut of L-shaped cantiever with oad in NC Figure 5.7: Interna forces resut of L-shaped cantiever with oad in NC Figure 5.8: Stress resut of L-shaped cantiever with oad in NC 19

21 Tabe 5.2: Comparison between theory and SCIA for L-shaped cantiever with oad in NC Expectation Resut in SCIA u y mm mm u z mm mm ϕ x mrad 0 M x Nmm 0 τ max 5.98 N/mm N/mm Point oad in shear centre The next oading scenario that wi be addressed is a scenario where a point oad is appied at the shear centre of the cross section. According to the theory, no torsion wi occur. This means the structure wi efect in y- and z-direction, but no axia deformation wi be present. An schematic view of the structure is presented in Figure 5.9. Figure 5.9: Cantiever beam oaded with a point oad in the shear centre The shear force centre is indicated with a cross in the drawing of the cross section. The ocation of this shear centre has aready been determined. The eccentricity of this point with respect to the norma force centre is d y =59.21 mm and d z =59.21 mm. For the cacuation process of this ocation, reference is made to Appendix C.1. Because the force has the same magnitude as in the previous exampe, and the dimensions of the structure have not changed, the defections due to bending moments remain unchanged. The ony difference is no rotation around the x-axis is to be expected. The expected defections are: [ ] [ ] uy mm = (5.7) u z mm In SCIA, the same structure is used as in the previous exampe. However, now the oad is changed to a point oad in the shear centre. Because there is no option to do this directy, the eccentricity of the oad is inputted manuay. The used offsets are d y =59.21 mm and d z =59.21 mm. The resuts from the cacuation are presented in Figure 5.10, 5.11, 5.12 and A resuts are coected in tabe 5.3. The first observation from this tabe is that the magnitude of the defection in y-direction is according to our expectation. The vaue for defection in z- direction however, is not the same. But possiby the biggest discrepancy is that SCIA reports axia deformation around the x-axis. This is against our expectation. We aso see an interna torsiona moment in the resuts. The shear stresses that are reported due to shear centre eccentricity, again have a vaue of 8.49 N/mm 2. 20

22 Figure 5.10: 3D-view of defected L-shaped cantiever oaded in the shear centre in SCIA Figure 5.11: Tabe of defections of L-shaped cantiever oaded in the shear centre in SCIA Figure 5.12: Interna forces resuts from SCIA Figure 5.13: Stress resuts with option consider torsion due to shear centre eccentricity enabed Figure 5.14: Stress resuts with option consider torsion due to shear centre eccentricity not enabed 21

23 Tabe 5.3: Comparison between theory and SCIA for L-shaped cantiever with oad in shear centre 5.4 Findings Expectation Resut in SCIA u y mm mm u z mm mm ϕ x 0 mrad 14.8 M x Nmm τ max N/mm 2 The anaysis of different oad scenarios has ed to some unexpected resuts. Athough the cacuations for pure torsion(section 4.1), and pure bending(section 5.1) both are in ine with the theory, it seems the inconsistencies start when bending and torsion are combined. In section 5.2 we observe that no torsiona deformation and interna torsiona moment are cacuated. We aso observe shear stresses that have a different magnitude than we woud expect. Athough the reported stresses are bigger than expected and thus do not cause an unsafe situation, it raises questions on SCIA s soution strategy. 22

24 Chapter 6 Concusion Recaing from the introduction chapter, severa questions were raised regarding a torsion cacuation in SCIA Engineer. Foowing from a suspicion that was present regarding torsion cacuations for non-symmetrica cross sections, a structure was considered where shear centre eccentricity payed a significant roe in the stress cacuations. The magnitudes of the shear stresses found in a doube camped structure oaded in bending and torsion were 60% ower than the stresses that were expected by using a theoretica approach. In engineering practices this discrepancy can ead to very unsafe situations, so the issue shoud be addressed. This ed to an investigation on different aspects of the torsion cacuation, with the utimate goa to formuate an advice for the software deveopers. To accompish this, the area of interest has to be narrowed down. The investigation described in this paper consists of three sections. First the geometry was checked, then the boundary conditions were considered and asty different oading situations were appied. Beow, for each of these sections the resuts are presented and concusions are drawn. 6.1 Geometry The first check performed in this paper was whether the determination of cross sectiona properties was performed correcty in SCIA. This research was performed in Chapter 3. The question formuated in the introduction was: Does SCIA cacuate cross sectiona properties in the correct way? Two types of cross section were investigated, starting with a symmetrica I-shaped section. This type of cross section is very common in engineering appications, thus very interesting to consider first. The fact that this cross section is symmetrica in two axes, means the shear force centre coincides with the norma force center. This eft mosty the vaues for part I of SCIA s geometry cacuation, the vaues for bi-axia bending and axia force, to be checked. A comparison between the vaues from a hand cacuation and the vaues from SCIA was made, but yieded no discrepancies. Secondy, an L-shaped cross section was considered. In theory, this type of cross section is symmetrica in ony one axis. This means the position of the shear centre does not coincide with the norma force centre. This fact made this cross section very interesting to investigate, because forces appied in the norma force centre wi cause torsion to occur. When the vaues obtained by a hand cacuation were compared to the vaues from SCIA, most magnitudes matched. Ony the determination of the position of the shear centre yieded different resuts, as presented in tabe 6.1: A discrepancy in the cacuation of the shear centre was observed between the theory and SCIA. The difference between the expectation and the observed vaue was 0.2 mm. However, since this 23

25 Eccentricity of shear centre in principe y-direction Expectation SCIA Difference mm mm 0.23 mm Tabe 6.1: Comparison between expectation and SCIA difference is very sma, it is not ikey to be the cause of the issue described in the introduction. Therefore the choice was made not to investigate this any further. 6.2 Boundary Conditions The next question that was raised regarded the boundary conditions. This question was: Do different boundary conditions infuence the torsion resuts in SCIA? The checks to formuate an answer to this question were performed in Chapter 4. Here two different boundary conditions were considered for a beam with an L-shaped cross section. First a singe camped cantiever beam was anaysed. A simpe torsiona oading scenario was chosen, and the resuts from a hand cacuation were compared to a cacuation in SCIA. This test was performed to determine whether for a basic torsiona probem, the shear stresses and torsiona deformation is computed according to our expectation. The resuts from a cacuation in SCIA matched the expected vaues, so this confirmed there are no issues with the cacuation of torsiona deformation and shear stresses for a basic torsiona probem. Now a doube camped beam is considered. This check determines whether the distribution of the interna torsiona moment over the beam is correct. Again a non-symmetrica L-shaped cross section was chosen. The structure was anaysed for three different positions of the oad, simiar to the exampe from the introduction. However, now the oad was chosen as a pure torsiona moment. The resuts from a cacuation in SCIA showed no discrepancies compared to the expected vaues. Therefore we can state that no issues are present in the distribution of interna torsiona moments. For both the singe-camped situation and the doube-camped situation, the cacuation in SCIA was correct. Therefore it can be concuded that the boundary conditions have no infuence on the resuts of torsion cacuations in SCIA. 6.3 Loading The third question that was formuated in the introduction was meant to investigate the infuence of different oading situations on the resuts for torsion cacuations in SCIA. The question was: Is the discrepancy between the theory and SCIA infuenced by varying the oading? In Chapter 5, three different oading scenarios were considered for the same structure. The structure consisted of a singe-camped L-shaped cantiever. The first check that was performed was if the defections of a non-symmetrica cross section oaded in bending ony were considered, the vaues of a hand cacuation matched the resuts from SCIA. The structure was anaysed with a hand cacuation first. Comparing the resuts from this cacuation to the resuts from SCIA, no differences were found. This means SCIA does use the correct soution strategy for non-symmetrica cross sections oaded in bending. The second oad scenario that was appied consisted of a point oad in the norma force centre. It was previousy estabished that this oading type shoud cause torsiona deformation of the beam, due to shear centre eccentricity of the cross section. However, observing the resuts obtained from SCIA, no torsiona deformation was reported. Aso no interna moment around the 24

26 x-axis was seen, athough this was expected. When the resuts for shear stresses were checked, another inconsistency was found. The shear stress did not have the expected magnitude. Athough an option is present in the program to consider torsion due to shear centre eccentricity, incorrect shear stresses were found. The stresses reported with this option enabed were higher than expected. The fina oading scenario that was considered, was simiar to the previous scenario. However, the point of appication of the force was now moved to the shear centre. According to the theory, this oading situation woud cause pure bending of the structure. However, in this case SCIA does report torsiona deformation. Aso an interna moment around the x-axis, and shear stresses due to torsion were reported. The difference between the shear stresses with the option consider torsion due to shear centre eccentricity enabed and disabed was the same exact vaue as in the oad case where the oad was appied in the norma force centre. Combining the knowedge gathered from these three checks, some concusions can be drawn about the infuence of oading situation on torsion resuts. The first is that for a beam oaded in pure bending or pure torsion, the vaues in SCIA match the expectation. However, when a oad case is considered where torsion due to shear centre eccentricity is present, the resuts for the torsiona deformation are not correct. In these cases aso the interna torsiona moment is not reported correcty. The option consider torsion due to shear centre eccentricity dispays shear stresses in non-symmetrica cross sections oaded in the norma force centre. However, the magnitude of theses stresses is not correct. 25

27 Chapter 7 Recommendations Stiffness Matrix It has been estabished in this report that the probems regarding torsion in SCIA engineer ony occur when shear centre eccentricity is present. Cacuations on pure bending and pure torsion produce correct resuts when compared to the theory. Ony when a oad is appied in the norma force centre of a non-symmetrica cross section, no torsiona deformation and interna torsiona moment in the structure are reported. This is not in ine with the theory. Given these symptoms, an issue may be present in the cacuation of beam eement stiffness for the stiffness matrix. If shear centre eccentricity in this matrix is not impemented correcty or even missing competey, this may expain the unexpected resuts. Perhaps the soution is very simpe, and the probem can be eiminated by adding the shear centre eccentricity to the offset that an eement norma centre ine may have with respect to the ine drawn in the mode. Shear stresses The resuts from the research in this paper indicate that SCIA cacuates the shear stresses in post-processing, which is after the output fie from the kerne is imported into the Graphica User Interface. In the 3D-stress resuts section, SCIA gives the option consider torsion due to shear centre eccentricity. With this option enabed, shear stress resuts are dispayed. However, the magnitude of these shear stresses differ from their theoretica expectation. This issue was presented in the introduction of this report, in section 1.2. When the stiffness matrix is corrected, the shear stresses due to torsion wi be computed directy from the interna torsiona moments. This means the vaue wi be correct, independent of shear centre eccentricity. Therefore this option and the underying code can be removed. 26

28 Bibiography [1] Coenraad Hartsuijker, Toegepaste Mechanica - dee 2: spanningen, vervormingen, verpaatsingen, Academic Service, Schoonhoven, 2nd edition, [2] Hans Weeman, Introduction into Continuum Mechanics, TU Deft, December 2007 [3] Hans Weeman, Lecture notes for Structura Mechanics 4 - Modue: Non-symmetrica and inhomogeneous cross sections, TU Deft, Apri

29 Appendices 28

30 Appendix A Torsion Theory In this paper, structures wi be discussed where torsion pays a roe. To provide some background on the the theory of torsion, this paragraph is presented to provide the reader with the necessary expanation of the used formuas. Coordinate system First a coordinate system is defined. We wi use the coordinate system pictured in Figure A.1. We observe a xyz-coordinate system where the x-axis coincides with the fibres in the norma Figure A.1: Definition of coordinate Axes force centre of the beam. This norma force centre, denoted by NC, is defined as The point in a cross section where the resutant norma force, caused by extension of the eement, acts [1, Chapter 3.1.3, p.69]. This aso means it is the point of the cross section where, if a norma force is appied, no bending moments wi occur in the eement. Aso a shear force centre can be defined. According to (Hartsuijker, 2003)[Chapter 5.5, p.331]: If the ine of action of the shear force goes through the point that is caed the shear force centre DC, no torsion wi occur. This point is denoted by DC in Figure A.1. In symmetrica cross sections, the shear force center coincides with the norma force centre. The z-axis is pointed downwards, so positive oads wi cause deformations in downward direction. Aso rotations have to be described. A rotation ϕ x around the x-axis is defined as a rotation from y to z. A ϕ y is defined as a rotation from z to x and a rotation ϕ z from x to y. A positive rotation is caused by a positive moment. The signs of the stresses aso foow these directions. Loading types To introduce the concept of torsion, we start with two simpe oading types in 2D. In Figure 29

31 Figure A.2: 2D-oading types A.2, two different oading situations are sketched. In the first case, the structure is oaded with a norma force. This wi cause deformation of the eement, in this case extension. The second exampe is oaded in bending. The oading causes bending moments to occur, which ead to the structure deforming in a curved shape. For some engineering purposes it wi suffice to ony consider these oading types for stress and strain cacuations. If we expand to 3D however, another oading type arises which needs to be addressed. This oading type is torsion, and is visuaised in Figure A.3. Torsion occurs when an eement is oaded with a moment around the beam axis. This type of oading causes axia deformation, which eads to a rotation of the cross section. Figure A.3: Torsion in a beam eement Stress and strain caused by torsion To anayse stresses and strains caused by torsion, in this paragraph three types of cross sections are presented. The first type of cross section that wi be discussed is the thin waed circuar cross section. The second type is a cosed circuar cross section. The third type that wi be discussed is a thin waed open cross section. for these three types, the formuas wi be presented for the cacuation of torsiona deformation and shear stresses. The theory presented here wi be used throughout the paper, which means this section is meant as a reference. The theory that is used is extracted from the book on engineering mechanics by Coenraad Hartsuijker[1]. For derivations and more information a reference is made to this book. Here ony the resuts of these derivations are presented, since proving the formuas is beyond the scope of this paper. 30

32 Thin waed circuar cross section Figure A.4: Exampe of thin waed circuar cross section In Figure A.4 an exampe of a thin waed circuar cross section is shown. As presented in (Hartsuijker, 2003)[1] in chapter 6.2 on page 373, the shear stresses for thin waed circuar cross sections can be assumed constant over the thickness of the materia. The vaue of this stress can be computed by equation (A.1). τ = M t R (A.1) I p Here, M t is the torsiona moment present in the cross section, R is the radius of the circe and I p is the poar moment of inertia. This moment of inertia has a vaue of I p = 2πR 3 t (A.2) Where R is the radius of the circe and t is the thickness of the materia. Subsequenty, a cross section oaded in torsion wi undergo torsiona deformation. This deformation is defined as a rotation around the x-axis, denoted by ϕ x. The constitutive reation between torsiona moment and deformation is given by χ = dϕ x dx = M t GI p (A.3) Where G is the materia specific torsion constant, and χ is the contortion of the beam. This contortion vaue shows arge simiarities to the ε in case of extension and κ in case of bending. For constant torsiona moment, the rotation due to this moment can be computed by ϕ x = M t GI p (A.4) Where is the ength of the eement. 31

33 Cosed circuar cross section Figure A.5: Exampe of cosed circuar cross section The next step is to expand to a cosed circuar cross section. An exampe of this type of cross section is presented in figure A.5. This type of cross section is thought to be buit up out of mutipe thin-waed sections fitted together. The constitutive reation from equation (A.3) sti hods, but the poar moment of inertia now has a different vaue, given by (A.5). I p = 1 2 πr4 (A.5) The assumption of constant stress over the thickness is not vaid anymore for this case. Now we observe a shear stress that is inear in the radius of the circe. This shear stress has a vaue of: τ(r) = M t r (A.6) I p Where r is the distance to the centre ine of the circe. Thin waed open cross sections An exampe of a thin waed open cross section is presented in figure A.6. It can be proven that Figure A.6: Exampe of thin waed open cross section for thin waed open cross sections, the constitutive reation for torsion deformation is of the form: ϕ x = M t (A.7) GI t where I t is now the torsiona moment of inertia. This is a different vaue than the poar moment of inertia we used before, and can be computed by: I t = n i= h it 3 i (A.8)

34 Which is the summation of the torsiona moments of inertia of every strip eement out of which the cross section is buit up. For each of these strip eements, h is the height and t is the thickness of the eement. For the cross section of the exampe in Figure A.6, this eads to a torsiona moment of inertia of I t = 1 3 ht bt3 (A.9) For thin waed open cross sections, the shear stresses are proportiona to the distance to the centre ine of the eements. The formua to cacuate the stresses is: τ = M t s c 1 2 I t (A.10) Where s c is the distance to the centre ine of the eement. For stress cacuations, we are mosty interested in the maximum shear stress. From this theory, we now know the maximum shear stress occurs when s c = t 2. With this knowedge we can write an expression for the maximum shear stress: τ max = M t t (A.11) I t 33

35 Appendix B Framework Anaysis For the automated cacuation and anaysis of structures a ot of different software is avaiabe on the market. The SCIA Engineer software package that is being used throughout this paper is a Framework Anaysis program. It aso incorporates finite eement technoogy. To understand how the program works, it is important to ook at a the individua steps the software takes to get from an inputted structure to a compete structura anaysis. To iustrate how this is done, in this chapter an overview wi be presented of the key steps to get from input to a cacuation, and utimatey to a usabe resut. Because this woud get very compicated very fast if we woud discuss every singe aspect of the process, we wi use a simpification of reaity. Athough this means we wi not discuss some processes in the program, it wi iustrate the operation of the software in a cean and simpe manner. B.1 Overview A fowchart of the steps necessary to get to a cacuation is pictured in Figure B.1. The process can be roughy divided into three steps. The first step is the generation of an input fie for the kerne. In this fie, for each eement in the framework, a stiffness matrix is generated. Aso the materia properties are imported from the GUI and the oads are processed. The second step is the actua cacuation. This is done in the kerne, which is the mathematica heart of the software. Here a arge K-matrix is generated, in which the stiffness matrices for a the individua frame eements are assembed. This eads to a arge system of equations, that can be soved for a the defections. These defections are then substituted back into the individua stiffness reations, to obtain the interna forces. The third an ast step, is importing these vaues in the output fie back into the graphica user interface. During this fina step the unity checks are performed and aso the stresses in the cross sections can be computed. Each of the steps wi be further eaborated in the foowing paragraphs. B.2 Generation of input fie The first step in any frame anaysis software is the generation of the input fie. The structure that has been created by the user has to be converted into a mode that can be evauated. This means the structure in the graphica interface first has to be divided into individua eements. To iustrate how this is done, we wi consider a simpe porta structure as pictured in Figure B.2 The eements are have a rigid connection in node B and C. The structure is oaded with a unit moment T in node B. The two supports in A and D are hinges, and a third support in C is added to prevent swaying of the structure. This means a the nodes can ony rotate, no atera dispacements are possibe. To start the modeing of this structure, we first divide the structure into three individua, static determinate beams. The mode becomes Figure B.3. Then we consider the beam between A and B. We are ooking for a system of equations in the 34

36 Figure B.1: Schematic overview of framework software 35

37 Figure B.2: Porta Structure form: m = K ϕ [ ] [ ] [ ] MA kaa k = ab ϕa M B k ba k bb ϕ B (B.1) Where m is the oad vector, ϕ is the dispacement vector and K is the fexibiity matrix. In words this equation means we have to find reations between the moments in the nodes and the rotations of the eements connected to the nodes. These can be found by soving the fourth order differentia equation for bending. The soution for this simpe oad case is aready known and can be found in the forget-me-nots. To simpify things, we wi first set ϕ A to 0, whie setting ϕ B to 1. This eaves us with two equations. [ ] MA = M B [ ] kaa k ab k ba k bb [ 0 1] [ ] [ ] MA kab = M B k bb (B.2) This means if we find the moments in A and B due to a unit rotation in B, we obtain the vaues of k ab and k bb. The rotation in B due to a moment in B can be found in Figure B.4. However, we are interested in the moment due to a rotation. This reation can be obtained by rewriting the expression, knowing that ϕ B has a vaue of 1: ϕ B = M B 4EI = 1 M B = 4EI k bb = 4EI (B.3) We aso know from the forget-me-nots, that in this type of oading situation the moment in the support in A wi be haf the moment excited at B. This means: M A = M B 2 = 2EI k ab = 2EI (B.4) 36