Stiffness Matrix for Haunched Members With

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1 Eng. & Technology, Vol., Suppl. of No., 7 Stiffness Matrix for Haunched Mebers With Abbas AbdelMajid Allawi* Received on: /9/4 Accepted on: // Abstract This study includes the derivation of the stiffness atrix for a haunched eber using the siple bending theory. The derived stiffness atrix covers ost possible geoetric shapes for haunched ebers under different loading cases and cobinations with including transverse deforations. The iportance of the transverse deforation in haunched ebers with high depth to span ratios is shown using nuerical exaple. The accuracy of the proposed analysis technique is verified by coparing the results of the nuerical exaple with those obtained fro the general analysis progra SAP9 using a large nuber of subeleents. مصفوفة الجساءة للاعضاء ذات النهايات المثخنة متضمنة ادخال تا ثير تشوهات القص العرضي الخلاصة هذه الدراسة تتضمن اشتقاق مصفوفة الجساءة للاعضاء الانشاي ية المثخنة عند النهايات استنادا على نظرية الانحناء البسيطة.مصفوفة الجساءة التي تم اشتقاقها تغطي معظم الاشكال الهندسية لهذه العتبات تحت تا ثير مختلف حالات التحميل والتداخل في الاحمال مع ادخال تا ثير تشوهات القص العرضي. ان اهمية تاثير تشوهات القص العرضي لهذه الاعضاء ذات نسبة عمق الى فضاء عالية قد جرى توضيحها في مثال عددي. وجرى اختبار دقة طريقة التحليل المقترحة بمقارنة النتاي ج للمثال العددي مع تلك المستحصلة من نتاي ج التحليل باستخدام برنامج التحليل SAP9 باستخدام عدد كبير من العناصر. ntroduction Haunched ebers can be used to shape the ebers in accordance with the distribution of the internal stress. By using these types of ebers, one can achieve the required strength with the iniu weight and aterial and also ay satisfy architectural or functional requireents. n industrial buildings, bridges, and high rise buildings, nonprisatic ebers with variable depth or width are usually used. Different approaches have been developed for the analysis of non prisatic ebers (ncluding haunched ebers). Reynolds and Steedan (988) published tables and graphs for analysis purposes. Siilar calculations are also given in other textbooks, for exaple by Tioshenko and Young (96), Vanderbilt (978) and Funk and Wang (988) calculated the stiffness atrix and fixed end forces by dividing the nonprisatic eber into subeleents. A refined analysis can be perfored by deriving the stiffness atrix and fixed end forces by 4

2 Eng. & Technology, Vol., Suppl. of No., 7 considering the exact variations of the geoetry. AlGahtani (996) derived the stiffness atrix by using differential equations and deterined fixed end forces for distributed and concentrated eber loads. Tioshenko and Young (96) concluded that if the variation of the cross section of a nonprisatic eber is not too rapid, it can be analyzed with sufficient accuracy by using the prisatic bea equations. AlMezani and Balkaya (99) deonstrated the probles due to discontinuity of eber axis and arching in analyzing nonprisatic ebers. The use of tables and graphs is liited to certain cross sections and span loads and is also difficult if several loading ay be considered. n the case of coputer applications, dividing the ebers into subeleents increases the nuber of equations and requires a larger aount of input data. Therefore, ost of latest studies have focused on a stiffness forulation of nonprisatic ebers, which considered the exact variation of the geoetry. The of deforations on fixed end forces has not been considered so far for nonprisatic ebers. The purpose of this study is to develop an exact solution using the siple bending theory for nonprisatic ebers with a wide range of span load variations and finite eleent forulation. Stiffness Matrix: The haunched ebers with a rectangular cross section and length as shown in Fig. is assued to be ade of hoogeneous, isotropic and linearly elastic aterials. Stiffness Matrix: The haunched ebers with a rectangular cross section and length as shown in Fig. is assued to be ade of hoogeneous, isotropic and linearly elastic aterials. Meber end displaceents, U, forces F, and fixed end forces P are shown in Fig.. The displaceent ethod yields the following eber equilibriu equation: F K. U + P () where K is the general stiffness atrix of a nonprisatic eber. Based on the conjugate bea ethod (Norriss et. al. 98), K can be suggested to be as follows: EA n EA n ( ( ( ( ( Γ ( + ) Γ n n EA EA ( ( ( ( ( Γ ( Γ 4

3 Eng. & Technology, Vol., Suppl. of No., 7 where A o and o are the area and oent of inertia for the prisatic part of haunched eber respectively and E is the odulus of elasticity. The of the variation of the area is expressed by the coefficients n and the variation of the oent of inertia by the coefficients,, and.when the eber is prisatic, n, 4 and. The factors, Γ, Γ and Γ account for the. n the case of BernoulliEuler theory, Γ Γ Γ (no transverse ). However, especially for ebers with high depth to span ratios, deforations should be considered to increase the accuracy. The coefficients, and are deterined by using the conjugate bea ethod and n is derived fro the forcedeforation. Fig. a shows the corresponding forces and couples when P, U, and all other displaceents are zero in eq. (). M(/( is applied as load to the conjugate bea as shown in Fig. d. Fro the equilibriu at the i and j ends, the following equations are obtained: A + x( ( B + x ( ( ( x( ( Denoting the integral as: () (4) x, ( x (, ( () then equations () and (4) can be rewritten in the for: A + B +! (6) where A and B in this case. posing a unit displaceent U 6, siilar equations can be written in ters of and for the case of A and B. Denoting the atrix of the coefficients in eq.(6) by C then for U, and U 6 and for U, and U 6, the following systes are obtained: C, C This yield h det C det C (7) (8) + det C where det C ( ) (9) when P, and U and all other displaceents are zero in eq. (), an axial force of agnitude n EA o / is developed. Then the forcedeforation relation gives: 44

4 Eng. & Technology, Vol., Suppl. of No., 7 n EA EA( () Solving eq. () for the unknown n yields: n A 4 where (/ A( ) A () 4 () A( Deterination of the factors n, and of the basic stiffness coefficients is based on the evaluation of the integrals of eqs. () and (). The integrals,, and 4 for the selected eber types can be calculated for different haunch shapes according to the taper factors α and α j at ends(see Appendix (A)). A hunched eber is assued to be consisting of three segents as shown in Fig.. The oent of inertia and the area in the regions with length i and j are variable and in the region with length o are constant. The integrals,, and 4 represent the suation of these three coponents. ntegrals for the constant and linear variations can be perfored analytically, while for any curved variation, nuerical integration is suggested to be done. k k k, k k k k k k k () The stiffness atrix of the eber shown in Fig. 4 corresponding to the coordinates F and F is: ( k ( ( (4) Using the relation between the transverse ing deforations and applied ing forces, deflection due to unit transverse load at end i can be expressed as Popov (968): U α α G A( G 4 () where G is the odulus and α is the shape factor of the cross section for. The following notation is used: 4 ψ α G ( ) (6) Shear deforation can be added to the inverse of eq. (4), i.e., the flexibility atrix as: Stiffness Factors for Shear Effect: Neglecting axial deforations and partitioning the atrices into four x subatrices, the bea stiffness atrices can be represented respectively in the following siplified fors: ( + ψ ) ( ) k ( ) + ( ) ( ) + ( ) (7) ( + + ) ( ) 4

5 Eng. & Technology, Vol., Suppl. of No., 7 The stiffness atrix k by considering the transverse deforation can be obtained by inverting eq. (7). The atrix k can be obtained fro the equilibriu conditions of the eber. Repeating the outlined procedure for the cantilever, by considering joint i as fixed end and releasing joint j, k and k can be obtained siilarly. The ultiplication factors, Γ, Γ and Γ are given as follows: G (8) 4( + G 4( ) Γ { + } G (9) 4 ( ) Γ { + } () G 4 ( ) Γ { + } () G Fixed End Forces Due to Axial Forces: Fixed end forces for the axially loaded nonprisatic eber are derived using the flexibility ethod. Referring to Fig., static equilibriu yields: P + P Q () Choosing P as a redundant and loading the priary structure with the actual load Q, the displaceent of point i due to P is: P P 4 δ P () n A E / E and the displaceent of point i due to the actual force Q is : Q Q δ Q (4) n A E / E where A is the iniu area along the length, n ( a) and A (/ A( ) (/ A( ) (b) Using copatibility at point i and the equilibriu condition (eq. ()), then: p 4 and Q 4 p 4 Q( ) 4 (6) (7) Fixed End Forces Due to Bending and Shear : The rest of the span loads considered are perpendicular to the eber axis, therefore P P 4. The force P and P 6 are calculated fro: P P 6 ji θi θ j (8) where θ i and θ j are the rotations at the ends of the eber due to the span load Q(. Using the conjugate bea ethod, these rotations, which are also given by Tioshenko and Young (96), can be expressed as: 46

6 Eng. & Technology, Vol., Suppl. of No., 7 θ θ i j E E M ( ( ( M( x ( (9) where M ( is the oent function due to span load Q( of the siply supported bea. The nuerical integration is used for the calculations of θ i and θ j in eq. (9). Adding rotations for the different loads and substituting into eq. (8) yields the fixed end forces P and P 6. Then P and P can be calculated by adding the respective siply supported bea and end rotations and the reactions due to the end oents. of the cross section due to transverse will produce additional deflection which is calculated by using the Bernoulli Euler bea theory. The nonprisatic eber shown in Fig. is subjected to eber forces Q(. Adding deforations over the length, the net deflection at end j with respect to origin i can be expressed as: U α V ( + V () G A( where V ( x ) is the function of force for the siply supported bea, which is obtained by differentiating the M ( function with respect to x, and V is the constant force due to end oents. Fro equilibriu, V [( + ) θ i + ( ) θ j Therefore, U using nuerical integration. can be calculated by ] () Additional fixed end forces caused by s are the fixed end forces that will prevent U in the bea. f the span load and the nonprisatic eber are syetric, then U and the additional fixed end forces will also equal to zero. Otherwise, a correction factor due to can be deterined fro eq. () and eq. () as follows: P P P P 6 ( + ) ( ) U ( + ) ( + ) () Finally, the vector P is deterined by assigning the corresponding coponents of the fixed end forces due to axial forces, bending and. Nuerical Exaple For the fixed ends haunched bea shown in Fig.6, the elastic odulus of concrete, E, is taken as GPa, the odulus. G, as GPa, the shape factor for,α, as. (for rectangular cross section), and eber width is.4. The bea is analyzed by using the proposed ethod for idspan heights, h, of.,.7,.,., and.. Fixed end oents at the left and right supports are deterined with and without the transverse deforations. The results are copared with that obtained using the wellknown finite eleent analysis progra SAP9. For SAP9 analysis, the bea is divided into 48 subeleents having constant area and oent of inertia defined at idsection of each eleent with and without considering the of transverse deforations. The result of analysis is shown in Table 47

7 Eng. & Technology, Vol., Suppl. of No., 7 A and Table B for both typed of analysis. When deforation is considered, the end oents increase. t is clear that the of deforations increases as the depth to span ratio increases. Conclusions: The derived stiffness atrix is general and applicable to siple bending theory, and cover a wide range of depth variation for haunched beas. The proposed forulation is also general and can be used for other types of nonprisatic or haunched ebers. Fixed end forces due to transverse deforations are considered, therefore ore accurate results can be obtained in the case of high depth to span ratios. 4Mebers with haunches can be analyzed as one eleent. This will reduce the nuber of equations, input data, tie and effort copared to the analysis ethod of dividing into prisatic subeleents, and that is very useful in frae analysis with nonprisatic ebers. The proposed eleent is convenient to use with the general displaceent ethod. t can be adapted to any finite eleents analysis progra. ist of Sybols : a subdivision length (Fig. ) A, B left and right support reaction A i, A j cross sectional area at i and j ends A( ross section area of the bea A o in. area of the eber cross section A in. area along the length C atrix of the coeff. in eq. (6) E odulus of elasticity F eber force vector F.. F 6 eber forces G odulus i, j oent of inertia at i and j ends ( oent of inertia at x fro origin o in. oent of inertia of eber cross section... 4 integrals given in eqs.() and () integral given in eq. (b) K.k 6 stiffness coeff. for U K, k bea stiffness with neglecting axial deforation K eber stiffness atrix length of eber i, j length in variable left and right haunches o length in the constant idsection. subdivision length M( oent function of the bea M ( oent function due to span load ( of siply supported bea n,,, factors of the basic stiffness coefficients n factor of the stiffness coeff. for length P eber fixed end 48

8 Eng. & Technology, Vol., Suppl. of No., 7 forces vector P..P 6 fixed end forces q( addition of q and q q,q M(/( function due to left right oent Q load intensity Q( span load function U eber displaceent vector U..U 6 eber disp. U deflection U total deflection of nonprisatic eber at right end V ( function of force for siply supported bea V constant force due to end oent a shape factor for a i a j taper factor at the left and right haunches b, G, G, G consideration factors total axial elongation in length δ Q disp. of point i due to the actual force Q δ Pi disp. of point i due to P θ, θ j rotations at the left and right supports of the siply support bea ψ factor defined in eq. (6) Structural Engineering, Vol., No. : 49. AlMezini,N. and Balkaya,C., 99. Analysis of Fraes with Non Prisatic Mebers, ASCE Journal of Structural Engineering, Vol. 7, No. 6: 79. Funk, R.R. and Wang, K.,988.Stiffness of NonPrisatic Mebers, ASCE Journal of Structural Engineering, Vol. 4, No. : Popov, E.P., 968. ntroduction to Mechanics of Solids. Prenice Hall, nc., Englwood Cliffs, New York. Reynolds, C.E. and Steedan, J.C,988. Reinforced Concrete Designer Handbook, th ed., ondon, E. & F.N. SPON. Tioshenko, S.P. and Young, D.H., 96. Theory of Structures. McGrawHill Book Co., nc., New York, N.Y. Norriss, C.H., Wilbur, J.B. and Utku, F.,98. Eleentary Structural Analysis. McGrawHill Book Co., nc., New York, N.Y. Vanderbilt, M.D., 978. Fixed End Action and Stiffness Matrices for NonPrisatic Beas. Journal of Aerican Concrete nstitute. Vol. 7, No. 7, pp References: AlGahtani,H.J., 996. Exact Stiffness Matrix for Tapered Mebers.ASCE Journal of 49

9 Eng. & Technology, Vol., Suppl. of No., 7 Appendix A : Depth Equations for Haunches : () Stepped Haunches : h(h i const. for x i h(h o const. for i x (i + o ) h(h o const. for ( i+ o ) x () inear Haunches: h(h i α i. for x h(h o const. for i x ( i + o ) h(h i αi ( for ( i+ o ) x () Parabolic Haunches : h( α i x / i α i x.+ h i for x i h(h o const. for i x ( i + o ) h( α i ( x ( j )x+( j ) / i + h o for ( i+ o) x eft support Right support Table A: Fixed end oents, kn. (Proposed Analysis) h,..7 With Shea 9 76 r..7 W/O With Shea r W/O Table B: Fixed end oents, kn. (Finite Eleent Analysis Using SAP9) eft support Right support h, With Shea r W/O With Shea r W/O

10 Eng. & Technology, Vol., Suppl. of No., 7

11 Eng. & Technology, Vol., Suppl. of No., 7 F F 6 F E, A(, ( i j F y Fig.4: Nonprisatic eber Subjected to F and F P Q P 4 x a Fig.: Axially loaded eber..... kn kn 7 kn. kn/ h kn/ h Fig.6: Nuerical exaple