Deflection of beams problems with solutions pdf. P-654, find the value of EI δ at 2 ft from R 2.
Deflection of beams problems with solutions pdf. The beam lengths are L2 = 6 m. Problem –1: Determine the deflection of a given beam at the point loads. This page titled 4: Solution Method for Beam Deflections is shared under a CC BY-NC-SA 4. Apr 4, 2020 · Solution to Problems | Chapter 9 | Deflection of Beams Textbook: Mechanics of Materials, 7th Edition, by Ferdinand Beer, E. SOLUTIONS. 3 Deflections using singularity functions or the Macaulay method W. As load is applied on a beam, it deflects. The degree of indeterminacy of the beam in examples 10. Sign Conventions for Beams In the analysis of beams, it is important to adhere to the generally agreed positive and negative signs for loads, shear forces, bending moments, slopes, and deflections. 4 The Moment Area Theorems. Method of superposition. Find the height h if the maximum deflection is not to exceed 10 mm. Solution 636. The slope-deflection method for beams will be illustrated using the example structure shown in Figure 9. Deflection Equation ( y y is positive downward) EIy = Px2 6 (3L − x) E I y = P x 2 6 ( 3 L − x) Case 2: Concentrated load at any point on the span of cantilever beam. In problems involving both bending and axial deformation, be careful with the units. ) Problem Set 2 (PDF) Solutions to Problem Set 2 (PDF) Development of Constitutive Equations for Continuum, Beams, and Plates. For the following beam, of dimensions 150 mmb = and 225 mmd = and E =10 kN/mm2, show that 71s0da r 4 θB =×− and 9. 3 Solution: 102 At x = 0, y = 0, therefore C2 = 0 At x = 4 m, y = 0 Therefore, At x = 2 m (midspan) Maximum midspan deflection Thus, 103 or Example 4. The shape may be superimposed on an x – y graph with the origin at the left end of the beam (before it is loaded). How to determin Apr 22, 2021 · To obtain the flexibility coefficients, use the beam-deflection tables to determine the support reactions of the beams in examples 10. 8 Virtual Work for Beams. The three-moment equation can be applied at any three points in any beam. Click here to show or hide the solution. Introduction. 4, determine (a) the deflection and slope under the load P and (b) the maximum deflection between the supports. y. 1 and 10. 1016/j. 37in ↓. Three sing le span bea ms are sug gested to. The values are given in tabular form with up to six significant figures. Christian Otto Mohr. Johnston, John DeWolf and David M A. Nov 1, 2022 · DOI: 10. Figure 1, the Jan 16, 2021 · Closed-form solutions for the coupled deflection of anisotropic Euler–Bernoulli composite beams with arbitrary boundary conditions January 2021 Thin-Walled Structures 161(2):107479 Boundary conditions: Deflection and slope at boundaries 2. v = deflection in the y direction. 3 Problems 1. Deflection of Beams Exercise Sheet Solutions - Free download as PDF File (. To develop the equations for the computation of deflection of beams and frames using the virtual work principles, consider the beam loaded as shown in Figure 8. the method using the differential equation which we have derived. Beam-Stiffness and moment carryover: to use for the analysis of statically indeterminate beams (unlikely that you get a SI frame). Mar 24, 2021 · The particular solution \(w_p\) of the beam deflection equation, Equation depends on the loading, but not the boundary conditions. the. GENERAL THEORY. Oct 28, 2015 · Table 2. M. eded with overhang, c) continuous beam, d) a cantilever beam, e) a beam fixed (or restrained) at the left end and simply supported near the other end (which has an overhang), f) beam fixed (or restrained) at both ends. various locations. 3. DEFLECTION OF BEAMS. Figure 5. 5 Practice Problems. 1. Determination of bending moment and shear force diagrams is the subject of elementary courses in statics, and the general procedure is not explained here. Figure E4. Problem 654. Problem –2: A steel cantilever beam of 6m long carries 2 point loads 15KN at the free end and 25KN at the distance of 2. The flexural rigidities of the beams are = 40,000 and E2I2 = 14,000 Determine the deflections of beam (1) at D and at B. edu 2019. A simple support for the real beam remains simple support for the conjugate beam. Problem 5-5: Continuity Condition. (This problem set corresponds to Lecture 3. The numerical technique used for evaluating the elliptic integrals is described. Of these methods, the first two are the ones that are commonly used. Integrate Moment-displacement differential equation. Maximum deflection. Sol'n: The bending moment in the beam is given by: M (x) = -P (L - x) ‹ Solution to Problem 648 | Deflection of Cantilever Beams up Solution to Problem 653 | Deflections in Simply Supported Beams › Add new comment 138927 reads At x L, y 0 9- 8 MECHANICS OF MATERIALS Sample Problem 9. x x dx EI W x dx (50 5) 1 ( ) 10 6 days ago · Choose formula: PL³/ (48EI). 6b, and Figure 8. 2 is 2. ) Problem Set 3 (PDF) Apr 16, 2021 · A cantilever beam shown in Figure 7. Equations for the calculations of the deflections of trusses and beams using the virtual work method. “design for stiffness”. Apr 16, 2021 · A cantilever beam is loaded with a uniformly distributed load of 4 kips/ft, as shown in Figure 7. Solve statically indeterminate beams: where the number of reactions at the supports exceeds the number of equilibrium equations available. 3 Integration of the Curvature Diagram to find Deflection. . Solution (M/EI) diagram. 0 license and was authored, remixed, and/or curated by Tomasz Wierzbicki (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 6a, Figure 8. We seek to nd conditions under which the beam will buckle, i. Slope Deflection Method: Slides from Leet et al. Req'd: Determine the deflection at the end of the beam. txt) or read online for free. P-654, find the value of EI δ at 2 ft from R 2. For the uniformly loaded beam the particular solution is the first term in Equation (4. Sivakumar Introduction The axis of a beam deflects from its initial position under action of applied forces. Then assuming a particular set of dimensions, the deflection and stress values of the beam are calculated analytically. 6 The Virtual Work Method. Compound beam. Example 7. For the beam of Example 3, using only Mohr’s First Theorem, show that the rotation at support B is equal in magnitude but not direction to that at A. Find the equation of the elastic curve for the simply supported beam subjected to the uniformly distributed load using the double integration method. SOLUTION hinge A B C The member is stable since the reactions are non-concurrent and nonparallel. 6c, respectively. 1 (Statically Determinate Beam) SOLUTION: • Develop an expression for M(x) and derive differential equation for elastic curve. • Integrate differential equation twice and apply boundary conditions to obtain elastic curve. B. Solve the problem of a simply-simply supported beam loaded by a point force acting at eh symmetry plane, but at a distance a from the left support. 4. C = v (L/2) = deflection at midpoint C of. 8. Apr 17, 2021 · Virtual Work Formulation for the Deflection and Slope of Beams and Frames. The free-body diagram for a beam ab having a Solving Problems of Simple Structural Mechanics - January 2022. Considering this assumptions at first using the Bernoulli-Euler's bending-moment curvature relationship, the approximate solutions of the cantilever beam was obtained from the general set of equations. Thus EI is an index of the bending (flexural) strength of an element – called Flexural Rigidity of the element. Sep 8, 2019 · beam load intensity, also the enhanced beam deflection problem is solved using two. L. Lecture Series. org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. This beam deflection calculator will help you determine the maximum beam deflection of simply-supported or cantilever beams subjected to simple load configurations. 2 The Bernoulli-Euler Beam Theory. Fig. Deflections and slopes of simple beams. 1 Introduction. Elastic Sathyabama Institute of Science and Technology Engr. Al-Tarafany. per ft. 12. 3 Using the slope deflection method, compute the end moments and plot the bending moment diagram. The deflection can be observed and measured directly. The flexural stiffness is 110 MNm2. Divide the beam into segments. In the unloaded condition, beam (1) touches but exerts no force on beam (2). Jan 1, 2013 · Request PDF | An analytical solution for the large deflection problem of Timoshenko beams under three-point bending | This article is conducted to analyze the large deflections of a straight in t he original beam will be equal t o t he shear force at A in t he conj ugat e beam , t hus, ( clockwise direct ion) The deflect ion of B in t he real beam will be equal t o t he bending m om ent at B in conj ugat e beam i. 6a. For the uniform beam, determine the reaction at A, derive the equation for the elastic curve, and determine the slope at A. For each beam shown below, draw the shear force diagram Sep 28, 2020 · Problems on simply supported beams. Include only the effects of bending in your virtual work equation (no axial work). The beam differential equation is integrated twice – deflection of beam at any c/s. 4th Ed. What is Macaulay's Method?2. The compound beam is unstable since the three reactions are all parallel. Example 2: Determine the deflection of the left end of the beam loaded and supported as shown in Figure, in terms of P, L, E, and I. II. Solutions to Problem Set 1 (PDF) The Concept of Stress, Generalized Stresses, and Equilibrium. Castiglino’s Theorem: When a body is elastically deflected by any combination of loads, the deflection at any point and in any direction is equal to the partial derivative of strain energy (computed with all loads acting) with respect to a load located at that point and acting in that direction. 1 4. Cantilever beam. It is also statically determinate. Enter values: 45 × 10³ × (4 × 10³)³/ (48 × 2. Macaulay first suggested that singularity functions could be of considerable use in solving problems on the deflection of beams. Maximum Moment. It is also indeterminate to the second degree. e. Dr. Calculate desired deflection (v) and slopes (θ) Module 5. 10b. To save this book to your Kindle, first ensure coreplatform@cambridge. For each beam shown below, determine the equations for the axial force, shear force and bending moment as a function of the position along the length of the beam. This information is contained in tutorial 2. Although the number of parts required for a specific task in compliant mechanisms can be significantly saved compared with that in rigid-body mechanisms, the high nonlinearity of the large static deflection of flexible beams leads to difficulty in 2. Tapered cantilever beam moment area method for beam deflections mechanics e method of superposition deflection of beam by unit load method puter aided deflection and slope. The beam is subjected to a compressive load P , as shown in the gure. 2: Determine the expressions for the de ection, slope, bending moment, and shear force for a beam clamped at both ends and subjected to uniformly distributed load of intensity q 0 and point load F 0 (upward) at the center. A cantilever beam is 6 m long and has a point load of 20 kN at the free end. 1 Work and Energy Work done by external forces on a material point or a structure is converted to internal work and internal stored energy. 36 mm δB = . Before Macaulay’s paper of 1919, the equation for the deflection of beams could not be found in closed form. EIABC = 2,000,000 k-in2 and EICDE = 800,000 k-in2. 2)mustbe zero. 664 by a uniformly distributed load of intensity wo acting over the middle half of the beam. Fig . • Locate point of zero slope or point of maximum deflection. 5. Compute the axial forces in the frame due to the virtual couple. the asymptotic solutions of larger numbers of terms. For the support movements shown, find the following: The vertical deflection at point E; The slope just to the left of the internal hinge at C; Show that, for the end loaded beam, of length L, simply supported at the left end and at a point L/4 out from there, the tip deflection under the load P is PL3 given by ∆= (316 ⁄ )⋅-----EI P A B C L/4 L The first thing we must do is determine the bending moment distribution as a function of x. 5 and draw the bending moment and shear force diagrams. The reactions are treated as part of Aug 12, 2012 · The paper's main contributions include the addition of an axial deflection model to existing beam bending models, the exploration of the deflection domain of a fixed-guided beam, and the Prof. In this paper, a comprehensive solution based on the elliptic integrals is proposed for solving large deflection problems. Continuity conditions: At a given point, the deflections (or slopes) obtained for the left- and right-hand parts should be equal 3. The cantilever beam shown in Fig. dx. No problem. Also, sketch the deflected shape of the beam. 6\). 37in ΔA − 0. Given: The rectangular beam, built in at the left end, having length, L, and cross-section of width, b, and height, h, is acted upon by a point load, P, at its free end. Area-moment method. Deflection at B. The terms 6Aa¯/L 6 A a ¯ / L and 6Ab¯/L 6 A b ¯ / L refer to the moment diagram by parts resulting from the simply supported loads between any two adjacent points described in (1). If the beam is designed based on the maximum allowable deflection, this is called. segments). USING ACI 318 PROCEDURE. Find the maximum deflection. Conservation of energy: 9. The procedure is as follows: Remove enough supports to make the problem statically determinate. mechmachtheory. Deflection by Integration. The deflection at the free end is 3 mm downwards. The left support reaction is 40kN and right support is 20kN. 2a, it is possible to observe that Procedure for Analysis. 2 Illustration of the Slope-Deflection Method Continuous beam with applied loads (deflected shape shown by dashed line) Figure 12. 28). 5 The Conjugate Beam Method. In the case of the simply supported beam with a point load Mar 1, 2021 · 1. Flexibility coefficients. Structural Analysis. The beam is simply supported over 8m with UDL of 10kN/m over the first 4m from the left support. Sivakumar Strength of Materials Deflection of beams Introduction Deflection of Beams (Solution Method by Direct Integration) Moment - Area Method for finding Beam Deflections Indian Institute of Technology Madras Strength of Materials Prof. Calculate the slope and deflection at the free end. For each truss below, determine the forces in all of the members marked with a checkmark ( ) using the method of sections. Sep 30, 2022 · EV ALUA TING SHORT AND LONG TERM DEFLECTIONS OF BEAMS. The load on the conjugate beam is the M/EI diagram of the loads on the actual beam. the beam (downward) x1 = distance from support A to. 4 × 10⁵ × 72 × 10⁶) = 3. Solution: For this problem, EIis a constant and q Mar 26, 2013 · The elliptic integral solution is often considered to be the most accurate method for analyzing large deflections of thin beams in compliant mechanisms. This structure is 4∘ 4 ∘ indeterminate, and so would be difficult to solve using the force method. of a new beam problem. Consider the following beam and its loadings. results (solutions) from c omplex AL Example 6. [1+ (dv=dx)2]3=2. A cantilever beam is 5 m long and has a point load of 50 kN at the free end. Real and virtual systems. The real and virtual systems are shown in Figure 8. 2a. boundary conditions to obtain elastic curve. S. A B A B C Deflection of beams. In this problem, the virtual moments are the real moments divided by 12 (from superposition). It will determine the relation among the moments at these points. \(Fig. 5m from the free end. 6) The functions have a somewhat unusual behaviour. x. Classification of structure. A fixed end for the real beam becomes free end for the conjugate beam. (Note that the beam is statically indeterminate to the first degree) SOLUTION: An example best demonstrates this method. 2 Illustration of the Slope-Deflection Method Free bodies of joints and beams (sign convention: Clockwise moment on the end of a member is positive) Figure 12. θ = Pa2 2EI θ = P a 2 2 E I. The length of a conjugate beam is always equal to the length of the actual beam. the beam can be in xstresses(showninFig. 7. First, draw the bending moment diagram for the beam and divide it by the flexural rigidity, EI, to obtain the diagram shown in Figure 7. ( downward direct ion) Ex a m ple 4 . EI is constant. 4 shows five singu larity functions of the set. By explicitly incorporating the number of inflection points and the sign of the end-moment load in the derivation, the For example, the length of the beam, as shown in Figure 1. Example : (Spring) (Ref Chapter 9) Example : (Trusses) (for conservative systems) (for linear spring) • There are multiple (infinitely many Jan 1, 2018 · An elastic analysis of continuous beams of equal spans is. 47 mm. Figure 4. 6. Use E = 10 GPa. 27-4. Dealing first with the two Numerical evaluations of elliptic integral solutions of some large deflection beam and frame problems are presented. We intend to find the deflection at mid-span of this beam. P-636 has a rectangular cross-section 50 mm wide by h mm high. 8. 1, is significantly greater than its breadth and depth. Use FBDs and equilibrium to find equations for the moment M(x) in each segment. Solution 654. Jan 1, 1981 · Numerical evaluations of elliptic integral solutions of some large deflection beam and frame problems are presented. 2022. The overhanging beam, from our previous example, has a fixed support at A, a roller support at C and an internal hinge at B. By using the method of superposition, we may determine the force imposed by a redundant support and use this information to supplement the equilibrium equations. 11. (This problem set corresponds to Lecture 4. ymax ≤ yallowable) To determine the reactions in statically indeterminate (SI) problems. Modulus of Elasticity = E Moment of Inertia = I. Replace each support with the reactions they exert. The heights h 1 and h Apr 16, 2021 · Thus, the deflection in the real beam at point A is as follows: ΔA = − 1728 ( 12)3 ( 29, 000) ( 280) = − 0. Here are the steps used to draw the conjugate beam from the real beam: Step 1: Draw the bending moment diagram for the real beam. This gives ˇdv=dxwhen the squared derivative in the denominator is small compared to 1. 7 Virtual Work for Trusses. Young’ s modulus of elasticity, E, is 30 E6 psi, and the 2nd moment of area, I, is 180 in. Direct integration method: The governing differential equation is defined as Sep 25, 2020 · Deflection Of Beams Problems With Solutions. A and B about B. Note : This method does not give us an expression/equation for the slope or deflection at ALL points of the beam (as required by the general Problem statement of Structural Analysis), whereas the method of double integration does. Support reactions. Elastic curve. Maximum deflection of the beam: Design specifications of a beam will generally include a maximum Problem 655 Find the value of EIδ under each concentrated load of the beam shown in Fig. Aug 18, 2023 · intuitive for the evaluation of solutio ns in comparison with. Direct integration method: The governing differential equation is defined as Beam (1) is supported by a fixed support at A and by a simply supported beam (2) at D. 3. The area-moment method of determining the deflection at any specified point. To determine the slope at free end & also deflection at free end I = 1 Aug 24, 2023 · Using the moment-area method, determine the slope at the free end of the beam and the deflection at the free end of the beam. Determine V and M relations for the beam. Oct 9, 2006 · The problem of large deflections of cantilever beams made of materials obeying a Ludwick type stress–strain law under the combined action of one vertical concentrated force at the free end and a Consider a cantilever beam of length L made of a material with Young's modulus E and whose uniform cross section has a moment of inertia with respect to the x 2 axis I22. Thiscanbeexpressedas. 10. In the notes of lecture 5 the solution of this problem was outlined, but not completed, Complete the derivation by calculating all four integration constants. Methods for finding the deflection: The deflection of the loaded beam can be obtained various methods. Examining the deflection shape of Fig. Example Problem. Using the conjugate beam method, determine the slope at support A and the deflection under the concentrated load of the simply supported beam at B shown in Figure 7. 7 Practice Problems. AREA‐MOMENT METHOD. Civil Engineering Department, Al -Nahrain University, Baghdad, Iraq, dhiaa. Chapter 12 1 §12. The beam has constant EI for both the spans. Selected Problem Answers. The compound beam is stable. Deflection:∆=dU dQ. 7a Selected Problem Answers. Beams are always loaded in the longitudinal direction. Take I = 64x10-4 mm4 & its Young’s modulusN/mm (E). conducted to determine the bending moment and deflection at. From this equation, any deflection of interest can be found. Michael Thomas Rex, National Engineering College, Kovilpatti, Tamil Nadu, INDIAThis video lecture explains1. 9 Virtual Work for Frames. Draw a FBD including reaction forces. m. With the cantilever beam under a point load at the tip in. Strain-energy method (Castigliano's Theorem) Conjugate-beam method. P-655. 2. First, removing the loads \(P U = ∫L( P2 2EA + T2 2GJ + M2 2EI + V2fs 2GA)dx. Use these equations to draw the axial force diagram, shear force diagram, and bending moment diagram. For the beam in Fig. Nodes A and C are fixed and so do not have any degrees-of-freedom (DOFs). Careful it is the virtual force/moment time the actual displacement (FL/AE for trusses, and M/EI for beams). point of maximum deflection. We wish to determine the deflection caused by a force F applied to the free end of the beam, at an angle θ from the horizontal. load, w Aug 24, 2023 · Fig. 2. When a beam bends it takes up various shapes such as that illustrated in figure 1. As an illustration, consider the same pin-pin supported beam loaded by the triangular line load Problem 665 Replace the concentrated load in Prob. H. \(EI\) = constant. successive derivatives of the deflection. (5. SA59: Calculating Slope and Deflection in Beams Using the Moment-Area Theorems . Nevertheless, one can find extremal values of slopes and deflections using this method, and usually these Aug 24, 2023 · Slope-deflection equations for mnd Moments: Modified slope-deflection equation when far end is supported by a roller or pin: Practice Problems. 1. 105033 Corpus ID: 251702219; Solutions to large beam-deflection problems by Taylor series and Padé approximant for compliant mechanisms @article{Wu2022SolutionsTL, title={Solutions to large beam-deflection problems by Taylor series and Pad{\'e} approximant for compliant mechanisms}, author={Ke Wu and Gang Zheng}, journal={Mechanism and Machine Theory}, year 3. 1 (continued) 3 §12. Solved Practice Problems Deflection Of Beams Notes If Chegg. 1) for a pin-pin supported and cantilever beam. 1 through Figure P11. EI = constant. 6 KN-m. The bending moment at each portion of the beam, with respect to the horizontal axis, are presented in Table 8. (Answers 0. It can solve both statically determinate and statically indeterminate beam problems. (positive upward) dv/dx = slope of the deflection curve. Compliant mechanisms composed of flexible beams have the potential advantages of high precision and reduced wear and backlash [1]. 1 General. Slope at end. 1 Using the slope-deflection method, compute the end moment of members of the beams shown in Figure P11. The deflection at point C due to the applied external loads is required. Select appropriate support, symmetry, and continuity conditions to solve for constants of integration. The deflection of the beam is needed for two main reasons: To limit the maximum deflection (i. 10\). D. pdf), Text File (. The deflection of B from the tangent at A (∆) is equal to the moment of the φ diagram between. along a beam is a semi graphical method utilizing the relations between. 00327 and -13 mm). Kenneth Alambra and Nicholas Swanson. 8 ft. Numerous methods are available for the determination of beam deflections. Macaulay’s Method is a means to find the equation that describes the deflected shape of a beam. The shear force diagram shows V reducing linearly from 40kN to 0kN over 0-4m and then constant at -20kN from 4-8m. Step 3: Draw the conjugate beam having the same length as a real beam. For each truss below, determine the forces in all of the truss members using the method of joints. These methods include: Double-integration method. ∆=∂U ∂Q. M = −Pa M = − P a. Solution (\(M/EI\)) diagram. Using the moment-area method, determine the slope at the free end of the beam and the deflection at the free end of the beam. 4 For the beam loaded as shown in Fig. Symmetry conditions: For example, the slope of the deflection curve at the midpoint is zero (for a symmetric beam under symmetric loads) Three rules for using moment--area method: Rule 1) Rule 2) Rule 3) The change in slope between A and B (θ) is equal to the area of the φ diagram between A and. 6 The Method of Sections. 17a. Table 8. Step 2: Divide the magnitudes of bending moments by flexural rigidity and draw the M/EI diagram. Bending moments at portions of the beam. Goal: Determine the deflection and slope at specified points of beams and shafts. We can then compute the bending moment of this new beam in which case its bending moment will represent the deflection of the original beam problem shown in Fig. This document is a written version of v ideo lecture SA59, which can be found Deflections using Energy Methods. 4 KN-m ; MFBA = +3. substitute 5. new transforms, which are complex AL-Tememe an d AL-Tememetransforms. E4. 4 This problem can be broken down into the following two problems: Case 1—Effect of 600 lb. Methods of Determining Beam Deflections. 4 ft. Up. The cross section of a beam can be rectangular, circular, or triangular, or it can be of what are referred to as standard sections, such as channels, tees, angles, and I-sections. 4. Using the method of singularity function, determine the equation of the elastic curve of the beam, the slope at the free end, and the deflection at the free end. These are the same as calculated in the previous problem: MFAB = -2. 8 Det erm ine t he deflect ion at t he free end of t he beam shown in Figure 4. theeban@nahrainuniv. 262 APPENDIX 1: EXACT ANALYTICAL SOLUTIONS OF STRAIGHT BEAMS Example A3. Slope at A. Consider a cantilevered circular beam as shown in Figure 5 that tapers from radius r1 to r2 over the length L. FBD and equilibrium for the entire beam → equations for reaction forces and moments. (a) Fixed end moments. 1The exact expression for curvature is d ds = d2v=dx2. Example 4. Write down the moment-curvature equation for each segment: EI v x ) = M ( x ) Integrate the moment-curvature equation twice → equations for v’(x Problem 636. Solution. 1 2 §12. Beam Slope And Deflection Table Er4 The 1 Source For Ering Tutorials. 10a is subjected to a concentrated moment at its free end. The one of the method for finding the deflection of the beam is the direct integration method, i. 1: The static boundary conditions for a full and half of a beam. However, we must first determine the total weight W acting on the beam in order to compute the resistance forces R 1 and R 2. mg jh aj lh ri uj le ct ob bj
Deflection of beams problems with solutions pdf. the beam (downward) x1 = distance from support A to.