Since b is simply supported, the bm at b is zero ie. This method considers the deflection as the primary unknowns, while the redundant forces were used in the force method. The slopedeflection method relies on the use of the slopedeflection equation, which relate the rotation of an element both rotation at the ends and rigid body. Analyze two span continuous beam abc by slope deflection method. The structure is thus reduced to a determinate structure.
In this method, if the slopes at the ends and the relative. Conclusion the general form of the fundamental equation of slopedeflection method is easily memorized, and the equations may be written for all members of a structure with little effort. The deflection is obtained by integrating the equation for the slope. Deflection in beams double integration method page 6 of 9 write an equation for moment for the right half of the beam using the free body diagram at the right. Slope ei dydx maximum deflection occurs at midspan and so the slope at this position will be zero. Able to analyze determinate beam deflection and slope by moment area method. Aug 09, 2017 in this video you will learn how to analyze a continuous beam having a support settlement using slope deflection method.
Differential equation of the elastic curve as shown, the vertical deflection of a, denoted by v, is considered to be positive if directed in the positive direction of the yaxis that is, upward in fig. The slope deflection equations for final moment at each end are 2 l 4ei 12 wl 2 l 2ei m f 1 l 2ei 12 wl 2 l 2ei m f b 2 ba ba b a b 2 ab ab a b in the above equations there is only one unknown b. B, write the equation of moment equilibrium at joint b 17 example 12. L ei l ei l ei m in which represents the sum of the bending stiffnesses of all.
If b is not a point of zero slope the equation gives the change of slope between a and b. Using momentdisplacement relations, moments are then known. Concept of fixed end moments obtained using unit load method 1 derivation of the slope deflection equation figure 12. Slope deflection equation i the slope deflection equation at the end a for member ab can be written as. Now, the required equation to solve for the rotation. An important characteristic of the slopedeflection method is that it does not become increasingly complicated to apply as the number of unknowns in the problem increases. Slope is equal to zero when x l2 substitute x for l2 gives. Cantilever beam concentrated load p at any point 2 pa 2 e i lei 2 3for0 px yax xa 6 ei 2. This beam theory is applied only for the laterally loaded beam without taking the shear deformation into account. Shear and moment diagrams for a continuous beam the slope deflection method is used to determine the shear and moment diagram for the beam shown below. The slope deflection method relies on the use of the slope deflection equation, which relate the rotation of an element both rotation at the ends and rigid body rotation to the total moments at either end. Note that in the solution, for spans ab and cd the shorthand slope deflection formula along with pinnedfixed fems are used.
An important characteristic of the slope deflection method is that it does not become increasingly complicated to apply as the number of unknowns in the problem increases. Example 1 determine the moments at b and d, then draw the moment diagram. Let us insert the values of c 1 and c 2 in slope equation and in deflection equation too and we will have the final equation of slope and also equation of deflection at any section of the loaded beam. However, the tables below cover most of the common cases. The slope deflection method was widely used for more than a decade until the moment distribution method was developed. Distribution methoddistribution method structural analysis by aslam kassimali theory of structures. The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by george a. Slope and deflection of beams mechanical engineering mcq questions and answers home category mechanical engineering mcq questions and answers slope and deflection of beams 1 a simply supported beam carries uniformly distributed load of 20 knm over the length of 5 m. Eiy boundary conditions give zero deflection when x 0. Displacement method of analysis slope deflection method. If you make the assumption to deflections are very small and that the slope of the elastic curve at any point is very small, the curvature can be approximated at any point by d2v dx2 m ei v is the deflection of the elastic curve the elastic curve 10 beam deflection by integration. Structural analysis of continuous beam by slope deflection. In this lesson, slopedeflection equations are derived considering the rotation of beam axis. The slopedeflection method is used to determine the shear and moment diagram for the beam shown below.
Use slope deflection equations to find the resultant end moments for the continuous beam shown in figure 74a. Introduction slopedeflection method is the second of the two classical methods presented in this course. Deflection method in displacement method,theunknown displacements are determined first by solving the structures equilibrium equations. Use the nowknown dof rotations to find the real end moments for each element of the beam sub the rotations back into the slope deflection equations.
Pdf the slopedeflection method is one of the classical displacement methods which is used to analyze the statically indeterminate beams. New modification for slope deflection equation in structural analysis article pdf available july 2015 with 3,057 reads how we measure reads. In the book, the theory and practice of modern framed structures, written by j. In the slope deflection method the individual equations are relatively easy to construct regardless of the number of unknowns. Write the slopedeflection equation for the members end moments in terms of unknown rotations. Wood page 19 of 26 additional slopedeflection insights. Basic idea of slope deflection method the basic idea of the slope deflection method is to write the equilibrium equations for each node in terms of the deflections and rotations. Deflection equation an overview sciencedirect topics. The new modified slope deflection equation is similar to the old modified slope deflection equation, but the only. Structural design and inspection deflection and slope of beams by dr. Slope deflection method study notes for civil engineering. What is the difference between slope deflection method and. Assume a and c are pinned and b and d are fixed connected.
Chapter5 deflection of beam page 10 from equation ii we get ei. Cantilever beam concentrated load p at the free end 2 pl 2 e i nm 2 3 px ylx 6 ei 24 3 max pl 3 e i max 2. Slope and deflection of beams mechanical engineering. Note that in the solution, in calculating the mba and mde, the shorthandmodified slope deflection formula is used. Objective to obtain slope and deflection of beam and frame structures using slopedeflection method 3. The solution of this equation is complicated because the bending. We can see the slope equation and deflection equation in following figure.
Objective to obtain slope and deflection of beam and frame structures using slopedeflection method 3 4. Slope deflection method and moment distribution method are both stiffness methods. Hibbeler, 7th edition, prentice hall structural analysis, hibbeler, 7th edition, prentice hall. The slopedeflection equations give us the moment at either end of each element within a structure as a function of both end rotations, the chord rotation, and the fixed end moments caused by the external loads between the nodes see section 9. Slope deflection method examples purdue engineering. For information on beam deflection, see our reference on. In this paper a new slope deflection equation is presented to simplify the solution for the end members with pin or roller supports at their ends even with external moments at these ends.
That is, node displacements are treated as the unknowns, after solving the stiffness equation for displacements, member forces and reactions are obtained. The ultimate goal is to find the end moments for each member in the structure as a function of all of the dofs associated with both ends. Use the resulting equilibrium equations to solve for the values or the unknown dof rotations solving a system of equations. The slopedeflection method relies on the use of the slopedeflection equation, which relate the rotation of an element both rotation at the ends and rigid body rotation to the total moments at either end. The given beam in figure 74a is statically indeterminate of degree 1. This tutorial goes through a slope deflection method. A number of analytical methods are available for determining the deflections of beams. New modification for slope deflection equation in structural analysis 388. Pdf new modification for slope deflection equation in. Concept of fixed end moments obtained using unit load method 1 derivation of the slopedeflection equation figure 12. Slope deflection method numerical example with support. Pdf new modification for slope deflection equation in structural.
Bending deflection differential equation method ae1108ii. The slopedeflection equations for the end moments are as follows. Determine the slope and deflection by using moment area method expected outcomes. Aerospace mechanics of materials aerospace structures. Note that in the solution, in calculating the mba and mde, the shorthand modified slope deflection formula is used. Sep 16, 2019 the forces moments are found using force displacement relations. The ultimate goal is to find the end moments for each member in the structure as a function of all of the dofs associated with both ends of the member. Cantilever beam concentrated load p at any point 2 pa 2 e i lei 2 3for0 px yax xa 6 ei 2 3for pa yxaaxl 6 ei 2 3.
In this video you will learn how to analyze a continuous beam having a support settlement using slope deflection method. Using the slope deflection equation the member end moments are. A comparison between results obtained from the hand solution and spslabspbeam software is provided to illustrate the features and capabilities of the spbeam and spslab software programs. Deflection equations when a continuous beam or a frame is subjected to external loads, internal moments generally develop at the ends of its individual members. Note that in the solution, for spans ab and cd the shorthand slopedeflection formula along with pinnedfixed fems are used. Derivation of the slopedeflection equation when the loads are applied to a frame or to a continuous beam, the member will develop end moments and become.
Determine the deflection and slope at point b in a prismatic beam due. Beam deflection formulae beam type slope at free end deflection at any section in terms of x maximum deflection 1. Shear and moment diagrams for a continuous beam the slopedeflection method is used to determine the shear and moment diagram for the beam shown below. Wood page 19 of 26 additional slope deflection insights. The tables below give equations for the deflection, slope, shear, and moment along straight beams for different end conditions and loadings. New modification for slope deflection equation in structural. In this case two spans ac and cd of the beam are to be considered. Integrating once more to get the slope equation gives. The methods of three moment equation, and consistent deformation method are represent the force method of structural analysis, the slope deflection method use displacements as unknowns, hence this method is the displacement method. In the slopedeflection method the individual equations are relatively easy to construct regardless of the number of unknowns. It is frequently possible to simplify the equations through noting where values of. You can find comprehensive tables in references such as gere, lindeburg, and shigley.
Solving continuous beam by slopedeflection equations. The forces moments are found using force displacement relations. Equilibrium equations end moments are expressed in terms of unknown rotation. In statically indeterminate structures, the beam axis rotates due to.