Slope and deflection of simply supported beam 3. This section covers shear force and bending moment in beams, shear and moment diagrams, stresses in beams, and a table of common beam deflection formulas. The beam is subjected to a point load of 40 kN and uniform distributed load of 15 kN/m. We will consider the beam deflection at the tip to be the sum of two parts: One part will be the deflection due to the beam acting as if it were cantilevered to a wall at the support point B, the middle figure below, and a second part due to the rotation of the beam at this imagined root of the cantilever at B —the figure left below. Includes reactions, shear V (x), moment M (x), slope θ (x), deflection y (x), and bending stress σ = Mc/I. Jul 16, 2025 · This calculator is for finding slope and deflection at a section of simply supported beam subjected to concentrated moment on both supports. Mar 29, 2022 · The deflection of cantilever and simply supported beams can be easily calculated using the double integration method or Vereschagin’s rule. Rules 1 and 2 are suitable when a slope is required. 17a. You also know the following: The deflection at the right end of the beam is 0. The actual rotation at fixed end beam supports should be taken as zero. Find the position and magnitude of the maximum deflection and show that the position is always approximately within of the centre. The span is 2 m and the point load is 200 kN at the middle. The deflection can be observed and measured directly. academy/ In this video, you will learn how to calculate the maximum slope and maximum deflection for a simply supported beam subjected to a central point load and a UDL. Mar 2, 2024 · A: The beam deflection and slope equations are derived under the following assumptions: the beam is simply supported, the material is linear elastic, the beam is subjected to a point load at the center, and the beam is subjected to small deflections. Apr 16, 2021 · A simply supported timber beam with a length of 8 ft will carry a distributed floor load of 500 lb/ft over its entire length, as shown Figure 7. We compute reactions, shear, bending moment, slope, and deflection, and show how to reproduce the results using AMESWEB calculators. This mechanics of materials tutorial goes over an example using the double integration method to find the deflection and slope of a simply supported beam with a point load. Additionally, it offers a step-by-step example to illustrate the calculation process and a downloadable A simply supported beam \ (AB\) carries a uniformly distributed load of 2 kips/ft over its length and a concentrated load of 10 kips in the middle of its span, as shown in Figure 7. Cantilever Beam – Concentrated load UNIT – I – SLOPE AND DEFLECTION OF BEAMS – SCIA1401 INTRODUCTION SLOPE OF A BEAM: slope at any section in a deflected beam is defined as the angle in radians which the tangent at the section makes with the original axis of the beam. 2 in. Question: Determine the slope at point A and the maximum deflection of the simply supported beam. Boundary conditions are applied to solve constants. Sep 13, 2025 · Based on Shape and Material Rectangular beam T-beam I-beam Steel beam Reinforced concrete beam Wooden beam Design of Simply Supported Beam The design of a simply supported beam involves ensuring it can carry applied loads without failure. Assume that From the information obtained on its curvature, we may get a fairly good idea of the shape of the deformed beam. 3b, is May 15, 2021 · Deflection of Beams when there is the vertical displacement at any point on the loaded beam, it is said to be deflection of beams. 5 2) + 10 (2) 8 A y Apr 6, 2024 · Beam equations for Resultant Forces, Shear Forces, Bending Moments and Deflection can be found for each beam case shown. ), bending moment (B. Apr 25, 2019 · In this video derive the expressions of deflection for simply supported beam with point load at mid position. The deflected shape diagram is used to identify relationships between vertical intercepts and rotations: This document discusses the deflection of beams under various loads. 72K subscribers 796 For instance, in the case of a simply supported beam with rigid supports, at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero. What will be the maximum deflection of elastic curve for a simply supported beam length (l) subjected to a uniformly distributed load w/m over the entire span? Deflection y = wl4 / 384EI 9. Deflection and Slope of Beams As load is applied on a beam, it deflects. Like and Subscribe. 5a. 5. It also includes 6 fundamental problems calculating slopes and deflections of cantilevered and simply supported beams. 7(a): Simply The document describes MATLAB code for drawing shear force (SF), bending moment (BM), slope, and deflection diagrams for a simply supported beam under a uniformly varying load (UVL). Select a beam and enter dimensions, then scroll down to see results. 6(f). Determine the slope at A and the maximum deflection. 7(b)). The deflection δ at some point B of a simply supported beam can be obtained by the following steps: 5. At this point, we can substitute all the values into the compatibility equations developed using force method. 5 Moment M'AB and MßA give final rotations 6A and 6B to the beam AB. We will also find the location of maximum deflection and its value by This calculator is for finding slope and deflection at a section of simply supported beam subjected to a concentrated moment on span. The general and standard equations for the deflection of beams is given below : Where, M = Bending For instance, in the case of a simply supported beam with rigid supports, at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero. L" | Find, read and cite all the research you need on ResearchGate Question: Determine the equation of the elastic curve, the slope at A, and the maximum deflection of the simply supported beam. . 1 Introduction in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along the axis of the beam Apr 16, 2021 · A simply supported beam is subjected to the combined loading shown in Figure 7. It happens due to the forces and loads being applied to the body. 3 mm when carrying a load of 200 kN at its mid-span position. They have calculated deflection and slope values but suspect unit errors may have led to unexpectedly small results How to use SkyCiv Beam Calculator Welcome to Beam Calculator, our free version of the SkyCiv Beam Analysis Software! Our calculator generates the reactions, shear force diagrams (SFD), bending moment diagrams (BMD), deflection, and stress of a cantilever beam or simply supported beam. 3 m from the free end? How to use SkyCiv Beam Calculator Welcome to Beam Calculator, our free version of the SkyCiv Beam Analysis Software! Our calculator generates the reactions, shear force diagrams (SFD), bending moment diagrams (BMD), deflection, and stress of a cantilever beam or simply supported beam. 72K subscribers 796 Graph the deflection function (or -w if you want your beam to sag down) over the interval [0,L] to see if your equation makes sense. The above beam design and deflection equations may be used with both imperial and metric units. Such a situation is impossible, because beam cross-section will go into plastic range, and the beam will no longer stay elastic. Link to other videos - Slope and deflection (max) of a doubly overhanging beam with point load and udl by Macaulay's Method. Compute reactions, shear force, bending moment, slope, deflection, and bending stress with unit conversion and plots. The maximum deflection of beams occurs where slope is zero. It simply means that the deviation from unsettling supports to the horizontal tangent is equal to the maximum deflection. at a built-in support, both slope and deflection are zero, whereas at a simple support, only deflection is zero (see Supports). For some applications beams must be stronger than required by maximum loads, to avoid unacceptable deflections. Mar 28, 2025 · If we know the relation expected between the load and the deflection, we can "back out" the material properties (specifically the modulus) from the measurement. (all slopes sum to zero and all deflections sum to zero). Jul 16, 2025 · This calculator is for finding the slope and deflection at a section of simply supported beam subjected to uniformly distributed load (UDL) on left-side portion of span. Jan 21, 2021 · The document provides an overview of simply supported beams, focusing on their deflection and slope under various load cases. It then provides equations to calculate the deflection and slope of simply supported beams with different load cases, including a central point load, point load anywhere on the span, uniform distributed load Slope and Deflection of Simply Supported Beam With Point Load by Macaulay's Method EduFriend E-Learning Platform 3. Determine the slope at end A. EI is constant. M. Q: How can these equations be used in practical applications? For the following simply-supported beam, we can find the rotation at A using Mohr’s Second Theorem. The deflected shape diagram is used to identify relationships between vertical intercepts and rotations: Question: 12-9. d. Since the deflection varies along the x-axis, v is a function of x. What is the Simply-Supported Beam Deflection Calculator? Definition: This calculator estimates the maximum deflection of a simply-supported beam under various load conditions, based on load, beam properties, and geometry. Following on from our previous article on , we’re going to look at deflection in a different type of beam. What is an en-castre beam? En-castre beams, or fixed beams, have fixed supports at both ends. What will be the max slope of elastic curve for a cantilever beam length (l) subjected to a point load w act at the free end? Slope ? = wl2 / 2EI 10. Mar 1, 2024 · Static analysis of a simply supported beam for point and distributed loads. Jul 16, 2025 · This calculator is for finding slope and deflection at a section of simply supported beam subjected to a point load. ), slope and deflection of a beam: deflection = y (or 6) Jul 27, 2021 · found from the types of support, e. F i g 7. slope of that deflection is the angle between the initial position and the deflected position. The load acting downward should be entered as negative whereas upward acting loads should be entered as positive. Determine the deflection at point C. Using the method of double integration, determine the slope at support \ (A\) and the deflection at a midpoint \ (C\) of the beam. 3 m from the free end? Jan 27, 2025 · The discussion focuses on calculating the slope and deflection of a simply supported beam for a mechanical college assignment. Note: For more information on shear, moment, slope and deflection calculations for different end constraints, please refer to "Beams; Flexure of Straight Bars" chapter of Roark's Formulas for Stress and Strain. ML ML It may be seen that rotation at loaded end is and at unloaded end is 3E1 6E1 Figure 1. Beams Deflection and Stress Formulas and Calculators Engineering Mathematics Double Integration Method Example Proof Simply Supported Beam of Length L with Concentrated Load at Mid Span The Double Integration Method, also known as Macaulay’s Method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve 1, and M 2 The slope and deflection of the right end of the beam due to V 2(two equations). In our previous article on , we looked at calculating deflection using standard tables. Prob. Free beam deflection calculators for simply supported, cantilever, and fixed-fixed beams. Strength and stiffness – design criteria for beams Strength criteria – SF & BM SLOPE AND DEFLECTION FOR A SIMPLY SUPPORTED WITH A UNIFORMLY DISTRIBUTED LOAD: A simply supported beam AB of length L and carrying a uniformly distributed load of w per unit length over the entire length is shown in fig. To find the rotations due to applied moment M in a beam without end rotation (Figure I. Related Calculations at ExcelCalcs Note! - deflection is often the limiting factor in beam design. Of particGlar importance is the knowledge of the maximum deflection of the beam. This document contains problems about determining bending stresses, slopes, deflections, and elastic curves of beams and shafts. kips by symmetry Figure 7. Simply Find the equation of the elastic curve for the simply supported beam subjected to the uniformly distributed load using the double integration method. Feb 15, 2019 · Slope and Deflection of simply supported beam with Point Load at Center by Double Integration Method WIT Solapur - Professional Learning Community 57. The deformed shape of the beam, called its elastic curve, is shown using dashed lines. A free, online tool to generate shear force diagrams, bending moment diagrams, deflection curves and slope curves for simply supported and cantilvered beams. Compute bending moment, shear force, slope and deflection for point loads, distributed loads, and moments. B rigid Jul 16, 2025 · This calculator is for finding slope and deflection at a section of simply supported beam subjected to a concentrated moment on left-side support. 14 is the modified slope-deflection equation when the far end is supported by a pin or roller. Assume EI is constant. Relationship Between Deflection, Slope and Radius of Curvature As we mentioned in our previous articles, a beam bends when it’s subject to A simply supported beam AB carries a uniformly distributed load of 2 kips/ft over its length and a concentrated load of 10 kips in the middle of its span, as shown in Figure 7. This calculator computes the bending moment, slope and deflection of a beam with both ends simply supported under action of a distributed load. In your case of a simply supported beam with a concentrated load at ANY point between the supports, the formula is for the deflection at the location of the load. The maximum deflection of such a simply supported beam of length 3 m is 4. The code then generates plots of SF, BM This video illustrates an example of applying conjugate beam method in determining slope and deflection of a simply supported beam subjected to a uniformly varying load. This calculator uses standard formulae for slope and eflection. Note that it is difficult to predict the direction (up or down) of the beam displacements due to the distributed load on the beam. 078 - Determine the slope and deflection of a simply supported beam with distributed and point loads Consider the beam and loading shown. Like 8. Jun 23, 2020 · In the case of simply supported beam there are two supports, a pin and a roller, both of which enforce zero deflection of the beam. For the following simply-supported beam, we can find the rotation at A using Mohr’s Second Theorem. 5) (8 4. Req'd: Determine the maximum deflection of the beam. Figure 1: A simply supported beam and its elastic curve Let’s denote the beam’s deflection as v. Nov 20, 2016 · We have provided illustrated solved examples on calculation of slope and deflection of cantilever, simply supported beams, truss and frames by diffferent methods like double integration, Macaulay's method, Conjugate Beam method, Moment area method and unit load method. Solution Support reactions. Problem Beam #3 The deflection at B and C is zero, as prescribed by the BCs. For a load not at mid span of the beam, the maximum deflection is not at the load point. A change of slope is called a curvature. Fig. Jan 22, 2025 · Solution For Find the slope and deflection of simply supported beam of span L, carryingi) Point load P at the centreii) a UDi of w kN/m over the entire span using the moment area method. Loads acting downward are taken as negative whereas upward loads are taken as positive. 3a. As with all calculations/formulas care must be taken to keep consistent units throughout with examples of units which should be adopted listed below: Given: A simply supported solid circular beam with radius r = 1. Many structures can be approximated as a straight beam or as a collection of straight beams. Engineers use formulas and building codes to calculate bending moments, shear forces, deflection, and stresses. Instruction for using deflection and slope calculator Sign Covention: All the distance should be measured with reference to origin taken at the extreme left end of the beam. 3\). Compute the maximum deflection. A simply supported beam is a type of beam that is supported at its two ends but is free to rotate and deflect under load. We can use the rules 1 and 3 to solve most problems, requiring evaluation of deflections. and length L = 50 in. \ (Fig. Beam bending is analyzed with the Euler-Bernoulli beam equation. For what position does the beam experience its maximum deflection? A beam's mode of deflection is primarily by bending, as loads produce reaction forces at the beam's support points and internal bending moments, shear, stresses, strains, and deflections. The most important characteristic of this beam is that the slope Feb 23, 2021 · https://engineers. It begins by defining deflection and slope, and describing the governing differential equation for beam deflection. If each beam has a Young's modulus of 200 GPa and a moment of inertia about its neutral axis of 46 106 mm4, determine the de ection at the centre G of beam CB. Jul 3, 2021 · #CivilSACSolved NumericalIn this, we will learn how to find the slope at a given point. As with all calculations/formulas care must be taken to keep consistent units throughout with examples of units which should be adopted listed below: Boundary conditions dictate deflection and slope values for simply supported and cantilever beams, crucial for calculations. This beam deflection calculator will help you determine the maximum beam deflection of simply-supported or cantilever beams subjected to simple load configurations. Use E = 235 GPa. 4K subscribers Subscribed Jun 4, 2023 · PDF | On Jun 4, 2023, Md Mubarak published DIRECT AREA METHOD SLOPE & DEFLECTION OF SIMPLY SUPPORTED BEAM CARRYING "U. g. Determine the deflection of en-castre beams. 7(a)), conjugate beam method may be used (Figure 1. Analyze and calculate shear force, moment diagrams, beam deflection, beam slope, beam reaction and free body diagram for simply supported beams and cantilvered beams. However, the analysis and design of a beam usually require more precise information on the deflection and the slope of the beam at various points. First simply supported beam which have supports at its both ends, second is overhanging beam which have length extended over The document provides an example of calculating the slope and deflection of a simply supported beam at point C using Macaulay's method. Question: Problem 09. The beam is 5 m long and has a point load of 1. Question: E = 200 Gpa and I = 45. Simply Supported Beam Deflection Formula with Central Point load Apr 16, 2021 · 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. Nov 27, 2011 · A simply supported beam of length carries a load at a distance from one end and from the one . The moment, slope and deflection equations are developed. Find the flexural stiffness of a simply supported beam which limits the deflection to 1 mm at the middle. 7. The spreadsheet determines and displays the MAXIMUM deflection in the span. Finally, the slope and deflection at Simply Supported Beam Deflection Example Worked example for a timber beam with multiple point loads and a uniform distributed load (UDL). Jan 17, 2021 · A simply supported beam rests on two supports (one end pinned and one end on roller support) and is free to move horizontally. is subjected to a uniform distributed load of q (x) = 24 lb f /in. It includes 6 preliminary problems analyzing internal bending moments and boundary conditions for different beam configurations. The material of beam has modulus of elasticity as 200 GPa. 12a. Counter-clockwise moments and rotations are taken as positive whereas clock-wise moments and rotations are Variables Used Deflection of Beam - (Measured in Meter) - Deflection of Beam Deflection is the movement of a beam or node from its original position. Slope of the beam is defined as the angle between the deflected beam to the actual beam at the same point. Determine the equation of the elastic curve for the simply supported beam using the x coordinate. Therefore, at these two points deflection is known and equal to 0. BEAM DEFLECTION FORMULAE BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF x MAXIMUM DEFLECTION 1. Bending moments, shear, deflections, slopes. of 600 N/m along its entire length. May 18, 2019 · Introduction to deflection of beam Load supporting element in any structure that is installed in horizontal direction and take load perpendicular to its length is called a beam. F. 9. End rotations due to M'BA. El ML 6E1 Slope Deflection Method Figure 1. In this article, however, our focus will be on calculating deflection using calculus. This calculator is for finding slope and deflection at a section of simply supported beam subjected to a concentrated moment on right-side support. Apr 6, 2021 · Hello Everyone, This Educational video is related to machanical Engineering subject Som. The code calculates the reactions forces, defines the BM equation, integrates to find slope and deflection equations. Beams are characterized by their manner of support, profile (shape of cross-section), equilibrium conditions, length, and material. The framework consists of two steel cantilevered beams CD and BA and a simply supported beam CB. May 3, 2024 · The tutorial provides beam deflection definition and equations/formulas for simply supported, cantilever, and fixed beams · Beam deflection calculator Jul 16, 2025 · This calculator is for finding the slope and deflection at a section of simply supported beam subjected to uniformly distributed load (UDL) on full span. Based on the way it is installed in structure, beam can be classified in to four main types. SIMPLY SUPPORTED BEAM (VARYING CROSS-SECTION)-SLOPE, LOCATION,MAXIMUM DEFLECTION, MOMENT AREA METHOD Deflection of Beams using Moment-Area Method - Intro to Structural Analysis In this video deflection problem solve by MACAULAY'S method, find slope and deflection of simply supported beam with several different masses. Using the method of singularity function, determine the slope at support A and the deflection at B. Hint: Maximum deflection occurs at point of zero slope. The remaining Apr 22, 2024 · This video explains how to find Slope & Deflection in case of a simply supported beam subjected to central point load with the help of a Macaulay's method. Let us consider the above-named methods with the examples below; Deflection of beams Goal: Determine the deflection and slope at specified points of beams and shafts Solve statically indeterminate beams: where the number of reactions at the supports exceeds the number of equilibrium equations available. In the case of a cantilever beam, there is only one These two constants must be evaluated from known conditions concerning the slope deflection at certain points of the beam. l. We will show, for instance, that the deflection at the midpoint of a beam subjected to "three-point bending" (beam loaded at its center and simply supported at its edges) is δ P = P 3 Calculate the flexural stiffness of a simply supported beam which will limit the deflection to 2 mm at the middle. [ 169 mm] Determine the vertical de ection at the end A of the bracket. For instance, in the case of a simply supported beam with rigid supports, at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero. Here, I'll outline a simple approach using the Euler-Bernoulli beam theory and numerical integration. 0 inches and the slope is zero radians. 2 kN at the middle and a u. For this reason, the analysis of stresses and deflections in a beam is an important and useful topic. Summary The following relationships exist between loading, shearing force (S. Aug 23, 2024 · Simply-Supported Beams Remember the following formulas for determining the maximum slope and deflection of simply supported beams under various types of loading. Handy calculators have been provided for both metric and imperial beam design and assessment. Figure 1 shows a simply supported beam subjected to a generalized load. Free simply supported beam calculator for a concentrated moment M applied at position a. This calculator uses standard formulae for slope and deflection. This calculator calculates the End Slopes, Support Reactions, Maximum Deflection and Maximum Stress in a simply supported beam with a point load. Simply supported beam. Using the moment area theorem, determine the slope at end B and the maximum deflection. Procedure for Analysis of Indeterminate Beams and Non-Sway Frames by the Slope-Deflection Method Apr 16, 2021 · A simply supported beam is subjected to the combined loading shown in Figure 7. The beam is made from 6061 aluminum. Units - choice of SI (fundamental), SI (mm), US Unit (ft) & US Unit (in). A jump in the slope gives an in curvature, and thus an in nite bending moments. This article will help you find the deflection and slope developed at any point of a simply supported beam, subjected to any load. It explains the methods to calculate these parameters, particularly using the double integration method, and includes formulas for different loading scenarios. 7(a): Simply Sep 25, 2024 · To calculate the deflection of a simply supported tapered beam under a uniform transverse load in MATLAB, you can follow a numerical approach such as the finite difference method or finite element method (FEM). The beam is made of material having a modulus of elasticity E. Closed-form formulas for simply supported beams with a point load, a linearly varying (trapezoidal) distributed load, and a point moment. Determine the slope at B and the deflection at point C of the simply supported beam using (i) double integration method, (ii) moment area method, and (iii) superposition method. The user has performed calculations using Macaulay's method but is uncertain about their accuracy due to a lack of access to simulation software. Aug 12, 2021 · This free online calculator is developed to provide a software tool for calculation of deflection and slope at any section "X" of simply supported beam (without overhangs) subjected to point load, uniformly distributed load, varying load and applied moments on the span or on the supports. So, they don’t allow horizontal, vertical or rotational movement of that support. Load per Unit Length - (Measured in Newton per Meter) - Load per Unit Length is the load distributed per unit meter. Applying the boundary conditions, the constants C1 and C2 are determined. This page provides a table listing deflection, slope, shear, and moment formulas for common configurations of beams. Find the maximum deflection. How do we analyse the deflection of simply supported beams using calculus. Nov 12, 2018 · Problem 6-1 Use Macaulay's method to determine the values of slope and deflection at 2m from the free end of the cantilever due to the imposed load as shown in figure 6-1 (a). May 1, 2021 · The tables below show beam deflection formulas for simply supported, fixed beam and cantilevers for different end conditions and loadings. To determine the reaction at support A of the beam, apply the equations of equilibrium, as follows: + ↶ ∑ M D = 0 26 (4. Plane sections normal to the beam axis remain plane and normal to the axis after deformation (no shear stress) Transverse deflection (deflection curve) is function of x only: ( ) v x Displacement in x-dir is function of x and y: ( , ) u x y du Question: Using the double integration method, derive the equations for slope and deflection for the simply supported beam in the figure below. Free online beam calculator tool for simply supported beams and cantilever beams. The slope and deflection of the right end of the beam due to M 2(two equations). • Slope and deflection (max) of a doubly over Why then slopes must be continuous for elastic beams? is simple. Apr 11, 2024 · In this section some examples will be given for the estimation of simply supported beam deflections and slopes, using the direct integration method. There are no constraints on the slope of the displacement at C. Jul 16, 2025 · This calculator is for finding the slope and deflection at a section of simply supported beam subjected to uniformly varying load (UVL) on full span. With the fixed condition at B, the slope at B is also zero. 5 Analysis of Indeterminate Beams The procedure for the analysis of indeterminate beams by the slope-deflection method is summarized below. 12-9 Feb 4, 2024 · The elastic curve slope of simply supported beam due to the actual loading needs to be calculated for both supports, namely at x=0 and x=L. Using the method of double integration, determine the slope at support A and the deflection at a midpoint C of the beam. Jul 16, 2025 · This calculator is for finding the slope and deflection at a section of simply supported beam subjected to uniformly distributed load (UDL) on a portion of span. Slope and Deflection of Simply Supported Beam With Point Load by Macaulay's Method EduFriend E-Learning Platform 3. The moment of inertia of segments AB and CD of the beam is I, while the moment of inertia of segment BC is 21 2 Feb 20, 2021 · Calculate bending moments, shear, slope and deflection of a simply supported beam at any point along its length when it is subject to a UDL (uniformly distributed load). V. In this video there is full derivation of simply supported Beam Carrying Mid point load find slope and In simply supported beams, the tangent drawn to the elastic curve at the point of maximum deflection is horizontal and parallel to the unloaded beam. 5 (10^6) mm^4. This method requires the calculation beforehand of the bending moment diagram. 11. Area constant and center of gravity of the bending moment diagram are essential for analyzing indeterminate beams. Equation 11. What would be the deflection at the free end of a cantilever of the same material, length and cross-section if it carries a load of l00 kN at a point 1.