neutral axis of t beam calculator

[Length]^3 A simply supported beam of overall depth 150 mm has neutral axis at 50 mm from extreme top fibre. , around y-y axis, which, for the T section, happens to be an axis of symmetry, the section modulus is found by the following formula: S_y = \frac{I_y}{X} \Rightarrow S_y = \frac{2 I_y}{b}. So if you want to calculate the moment of inertia of circle, moment of inertia of a rectangle or any other shapes, feel free to use the below software or our all-inclusive SkyCiv Section Builder. if(typeof ez_ad_units!='undefined'){ez_ad_units.push([[250,250],'mechcontent_com-large-mobile-banner-1','ezslot_1',162,'0','0'])};__ez_fad_position('div-gpt-ad-mechcontent_com-large-mobile-banner-1-0');Similarly, the bottommost fibre of the beam experiences the highest tensile stress and it gradually decreases as we move toward the neutral axis and it becomes zero at the neutral axis. The part of the slab which carries the load and connected with the beam as a single part of the L- beam or T- Beam is known as the flange of the T beam. While this discussion herein is limited to steel beams, the concept can be extended to other types of beam materials } The actual neutral axis of the given T-beam section can be determined in following two ways : When Compression in the Web is taken into Consideration : Considering Fig. find the area moment of each section by multiplying its area . Christian Science Monitor: a socially acceptable source among conservative Christians? OTHER CALCULATORS AREA MOMENTS OF INERTIA BEAM CALCULATORS TORSION OF BARS CIRCULAR FLAT PLATES BUCKLING ELASTIC CONTACT IMPACT LOADS NATURAL FREQUENCIES PRESSURED SHELLS 2) Uniaxial state of stress hypothesis: Any point on the beam is uniaxial stretched or compressed when deforming occurs regardless of the effect of uniaxial force on the sec-tion size. Whether you're working on a design project, conducting a survey, or studying for an exam, our Centroid Calculator is the perfect tool to help you get the job done. Multiply the number by the square of the radius. The elastic section modulus formula of a rectangle is S = bd/6, where: Take into account that this section modulus is about the horizontal centroidal axis. Calculate the udl it may carry if the max bending stress is not to exceed 100 N/mm2 . For the detailed terms of use click here. Steel reinforcement of area 1272 mm2 is placed at an effective depth of 400 mm. the failure will be on the compression face or on the tension face is applied around axis x-x, the section will respond with normal stresses, varying linearly with the distance from the neutral axis (which under elastic regime coincides to the centroidal x-x axis). Calculation Tools & Engineering Resources. , is the Young's modulus, a property of the material, and Key Properties, where the user inputs the geometry of their chosen section and the beam supports. Save my name, email, and website in this browser for the next time I comment. Input the details for the beam, then click the "Calculate Results" button: Structure Point Forces Dist Forces Constraints. How do I submit an offer to buy an expired domain? In USD (Ultimate Strength Design) as it used to be called in the USA or LRFD (Load and Resistance Factor Design) as it is now called or LSD (Limit States Design) as we call it in Canada, the neutral axis moves toward the compression side of the member. Additionally, its second moment of area about the horizontal centroidal axis is 22.1 in, while the elastic section modulus about the same axis is 7.31 in. in same units and this calculator will provide the "Output Results" in the y_{pna} =\left \{ \begin{array}{ll} h- \frac{A}{2t_w} & \text{ , if: } t_f \le {A\over2 b} \\ \frac{A}{2b} & \text{ , if: } t_f \gt {A\over2 b} \\ \end{array} \right. M_x Calculation Example - Calculate the Axial Forces of the Truss Members. In addition to its speed and accuracy, our Centroid Calculator is also incredibly easy to use. 3- Calculate excess moment (M 1) using Equation 15: M 1 =(300.23/.9) . The calculation of the polar moment of inertia Kids Bathroom Idea| The properties of the beam and section are specified by typing directly into the input fields. If your T section has widths from the bottom up w 1 = 3, w 2 = 1 and heights h 1 = 1, h 2 = 3 then the neutral axis is located from the bottom surface at a height of Lets start. To calculate the section modulus of a pipe pile of thickness t and radius R, use the section modulus formula for a very thin annulus: S = Rt, or follow these steps: Measure the radius R and the thickness t of the pipe pile. With a simple user interface, you can input your data and receive your results with minimal effort. To calculate the total moment of inertia of the section we need to use the "Parallel Axis Theorem" as defined below: Since we have split it into three rectangular parts, we must calculate the moment of inertia of each of these sections. The Moment of Inertia of a rectangle, or any shape for that matter, is technically a measurement of how much torque is required to accelerate the mass about an axis - hence the word inertia in its name. As the ratio I/cI/cI/c only depends on the geometric characteristics, we can define a new geometric property from it, called the section modulus, denoted by the SSS letter: As well as the second moment of area, this new geometric property is available in many tables and calculators, but if you want to know how to calculate section modulus from the moment of inertia, simply divide III by ccc, and you'll get it. if(typeof ez_ad_units!='undefined'){ez_ad_units.push([[250,250],'mechcontent_com-box-4','ezslot_9',106,'0','0'])};__ez_fad_position('div-gpt-ad-mechcontent_com-box-4-0');The uppermost fibres of the beam are under higher compressive stress and this compressive bending stress gradually decreases as we move towards the neutral axis and it becomes zero at the neutral axis. The position of the neutral axis from the bottom of the web is given by, `\bar{y}`=`\frac{A_{\text{web}}.y_{\text{web}}+A_{\text{flange}}.y_{\text{flange}}}{A_{\text{web}}+A_{\text{flange}}}`, Where,`A_{\text{Web}}` = Area of web`A_{\text{Flange}}`= Area of flange`y_{\text{Web}}`= Centroid of the web from bottom of web`A_{\text{Flange}}`= Centroid of the flange from bottom of web. Structure (Instructions) Specify the geometry and material of the beam: . At the certain location where the bending stress changes its nature from tensile to compressive, the value of the bending stress becomes zero and thus the fibres at this location never experience any bending stress. This calculator uses standard formulae and parallel axes theorem to calculate the values of moment of inertia as well as maximum and minimum values of section modulus about x-axis and y-axis of T section. Calculate the approximate steel requirements in relation. , around the x-x axis, is not the same for a top-end and a bottom-end fiber. font:normal 15px arial; Access our library of support articles to get started using ClearCalcs and learn advanced tricks to manage your projects and calculations. Multiplying the area of this element by its modules of elasticity, then by its distance from an arbitrary assumed reference axis. The neutral axis passes through the centroid of the cross-section, therefore by finding the location of the centroid we can locate the neutral axis. The final area, may be considered as the additive combination of A+B. This calculator is a versatile calculator and is designed to find the moment of inertia and centroid for any user-defined shape. Primitive Fireplace Decor| This calculator uses standard formulae and the theorem of parallel axes to determine moment of inertia. Radius of Gyration (Area): The capacity of a cross-section to resist bending. Find the moment of inertia of section I, rectangle, circle, triangle and all kinds of different shapes. Y_t if (!document.getElementById('elistaddress48375969659').value.match(regexPattern)) First, select the Cross-Section Type from the drop-down menu just below the diagram. The shear center is outside the channel web. if (e.innerHTML != "") return false; A_t At the bending stress apparatus there is a T beam on two supports and the clear span is 800 mm. Strength of Materials | Beam Deflection and Stress. Why are stress components on opposite faces identical? Although the material presented in this site has been thoroughly tested, it is not warranted to be free of errors or up-to-date. . Since M d <M u, the beam should be designed as a doubly reinforced concrete beam. In no way does thathipsterlife.com claim ownership or responsibility for such items, and you should seek legal consent for any use of such materials from its owner. Moment of Inertia Ixx (unit^4): Typically the most distant fiber is of interest. , the curvature of the beam due to the applied load. Point Loads are specified in units of force, kN or kip, and area applied at discrete points along the beam. We will take the datum or reference line from the bottom of the beam section. Let's say we have a rectangle with the following dimensions: Using the formula for moment of inertia, we can calculate the moment of inertia of the rectangle as follows: I = (b * h^3) / 12I = (2 * 4^3) / 12I = (2 * 64) / 12 = 10.67 in^4. Y_c As you can note, the transition from elastic to plastic is not uniform across the member, as some regions will reach the yield strength before others. It is important to note that the unit of measurement for b and h must be consistent (e.g., inches, millimeters, etc.). The Moment of inertia of section about neutral axis formula is defined as the quantity expressed by the body resisting angular acceleration. Now by simply finding the centroid of a composite section in a vertical direction, we can find the position of the neutral axis. Why engineers, architects and building designers depend on ClearCalcs to deliver their best work anywhere, anytime. The Reinforcement Beam Section Calculator is a failry simple tool, and is small part of our fully featured Reinforced Concrete Beam Design software offered by SkyCiv. The natural axis is the axis where there is no stress. The different start and end magnitudes must be specified by the user, and they can be used to represent triangular or trapezoidal loads. Start your free ClearCalcs trial to unlock saving and export, as well as even more calculators for wood, steel, and concrete beams, columns, and footings. Moment of Area of Large rectangle = M 1 = 62.56.25 = 390.25 cm. The location of neutral axis can be determined by applying the static equilibrium conditions. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. I_y border:0; In the Pern series, what are the "zebeedees"? A_c The cross-sectional area (see cross-sectional area calculator) above the neutral axis (between the inner surface of bending and the neutral axis) experiences compression stresses. The user inputs the name, magnitude and location from the left of the beam. A_t D. All the above Alternately, you can create your own custom section using our free moment of inertia calculator. SkyCiv Section Builder uses FEA to calculate key section properties, allowing you to model anything you need. The method to find a position of the neutral axis is different for simple beam and composite beam:-. For the case of a T section, under x-x bending, the position of the plastic neutral axis can be found by either one of the following two equations: \left \{ \begin{array}{ll} (h-y_{pna})t_w = \frac{A}{2} & \text{ , if } y_{pna} \ge t_f \\ y_{pna} b = \frac{A}{2} & \text{ , if } y_{pna} \lt t_f \\ \end{array} \right. Metals are not typically designed to go beyond the material's yield point. in compression, separated by the neutral axis. Thousands of engineers have used ClearCalcs to make design calculations the easiest part of their jobs. In Britain and Australia, these are typically reversed. LEAN INTO EVERY RIDE: Smart Lean Technology, keeps you confidently grounded and in command of the road with 6-axis Bosch IMU, Dynamic traction control, ABS with cornering control, and Drag Torque Control. This is a powerful tool for engineers, surveyors, designers, and students alike. The formula for calculating bending or normal stress on a beam section is the following: Where: M = internal bending moment about the section's neutral axis y = perpendicular distance from the neutral axis to a point on the section I = moment of inertia of the section area about the neutral axis The author or anyone else related with this site will not be liable for any loss or damage of any nature. When you do so, the input boxes below it will change to those required for the given type of cross-section. . Assumptions. t_f given below and click "Calculate". h The neutral axis is the axis through a beam where the stress is zero, that is there is neither compression nor tension. You can also use our section modulus calculator and do it faster. Kitchen And Bath Ideas| Any content, trademark/s, or other material that might be found on the thathipsterlife.com website that is not thathipsterlife.com property remains the copyright of its respective owner/s. The ratio of the width of the web to the depth of the web is kept as 1/3 to 2/3. Fix the overall dimensions of the T beam. Note: The module of elasticity of the steel section is 69000 N/mm2 A) Calculate the neutral axis of the T beam. Over neutral axis the stresses are zero. Loads, where the use can input distributed, point and applied moment loads. If that's not the case. Room Decorating App For Ipad| We can also write. Note that the derivation of bending moment equations above has the following assumptions: Firstly, the beam is linear and has a uniform cross-sectional area before stresses are applied. and Section Modulus Zxx (unit^3): In that case, the whole section is divided in two parts, one in tension and one in compression, each under uniform stress field. } } It allows you to: The moment of inertia calculator will accurately calculate a number of important section properties used in structural engineering. And thus it becomes clear that at the neutral axis the value of bending stress is zero. = y M / I (1d) where. } y_\textit{pna} Section modulus calculator for I beam, hollow rectangle, rectangle, C channel, T section, circular hollow section, round bar and unequal angle. Neutral axis locations for negative moment sections Sum of moments about the neutral axis as functions of kd: 1) Neutral axis within the slab (this case will govern for a very shallow beam web projection relative to the slab thickness) Left end negative moment section, f3((kd))b w+ + kd2 2 b fb w To find the neutral axis of any section including the T section in your question, ( called T beam because its cross section is a T): one picks a horizontal axis parallel to x axis (or the axis they need to find the beam's neutral axis). Therefore the position of the neutral axis for the rectangle is given by. section modulus about x-axis and y-axis of T section. indicates the neutral axis. A neutral axis is a line passing through the cross-section at which the fibres of the beam does not experience any longitudinal stress. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, I couldnot find anything that is why i asked here :). b-t+c at 5:30 t=0 and T1=39.3C at 5:35 t=5 min and T2=26.3C When it comes out of oven: Q: 2. font: normal 12px arial; corresponding units (unit2 , unit3 , unit4 We have a special article about finding the centroid of a compound area. A platform dedicated to engineering beams. It will help in deciding whether the failure will be on the compression face or on the tension face of the beam. regexPattern = "^[-!#$%&'*+./0-9=?A-Z^_`a-z{|}]+@[-!#$%&'*+/0-9=?A-Z^_`a-z{|}~. It will help in deciding whether Our paid account will show the full hand calculations of how the tool got to this result. The polar moment of inertia, describes the rigidity of a cross-section against torsional moment, likewise the planar moments of inertia described above, are related to flexural bending. Read more about us here. margin:auto; t_w in tension and margin-bottom:10px; The T beam section should be divided into smaller sections. beam is. For the section modulus where Neutral axis equation for geometric shapes: Why bending stress is zero at neutral axis? Radius of gyration Rg of a cross-section, relative to an axis, is given by the formula: where I the moment of inertia of the cross-section around the same axis and A its area. Step 1: Find actual neutral axis. A_c .bravenet-gobutton { We can derive the section modulus formula for a rectangular section by dividing its second moment of area (bh/12) by the maximum distance from the neutral axis (h/2): The elastic section modulus formula of a square is S = a/6, where: We can derive the section modulus formula for a square section by dividing its second moment of area (a/12) by the maximum distance from the neutral axis (a/2): To calculate the plastic section modulus of an I-section, use the formulas: Check out 34 similar materials and continuum mechanics , How to calculate section modulus from the moment of inertia, Plastic section modulus: beyond elastic section modulus, Section modulus formulas for a rectangular section and other shapes. It is defined as: where Keep visiting for getting updated or Join our mailing list, You may also visit links from our external sources, Instructions for Moment of Inertia Calculator, Deflection & Slope Calculator for Cantilever. It is directly related to the amount of material strength your section has. The moment of inertia of a tee section can be found, if the total area is divided into two, smaller ones, A, B, as shown in figure below. the offset from centroid, of a section fiber, parallel to x-x axis. We simply need to use the centroid equation for calculating the vertical (y) centroid of a multi-segment shape. { Chapter Reserve Problem 093 The cross section of pimetallic beam consists of an Jumirun bar (1) and copper bar (2_ pondej together The composite beam subjected to bending moment about the axis of Mz +150 N-m_ Gaowin below: The elastic modulus of the aluminum 72 GPa The astic modulus of the copper is Ez 106 GPa Assume 20 mm; C1 17 mm and C2 mm; The centroid of the equivalent al- Iuminum beam 13 . Solution of indeterminate structures slope deflection, moment distribution etc. The The Neutral Axis (NA) is located at the centroid. Does neutral axis always pass through centroid? Where S is section modulus I/Y. This relation is equivalent to the axial stress equation: =FA\sigma = \frac{F}{A}=AF. We can continue to calculate the moment of inertia. Greater than 4 kg/cm 2, but less than 20 kg/cm 2, shear reinforcement is provided. Moment of compressive area = Area in compression Distance between c.g. A measure of the ability of a cross-section shape to resist plastic bending, used to estimate the stress of a material when it begins to yield (reach its plastic limit) under a specific load. Once the loading and geometry have been specified, the calculator automatically uses the ClearCalcs finite element analysis engine to determine the moments, shear forces and deflections. Please enter the "Input Values" in the form background: white; Thomasville Furniture Raleigh Nc| display:block; Positive values imply a downward deflection and negative values imply an upward deflection. Multiply the width of section 2 by `n_{2}`. Refer below the calculator for more information on this topic, as well as links to other useful tools and features SkyCiv can offer you. Put simply, the Section Modulus is represented within a flexural stress calculation (such as in the design of beams) As you may know, we typically calculate flexural stress using the equation: The Elastic Section Modulus is represented in this equation as simply: After defining this, we can re-arrange our flexural stress formula as follows: There are two kinds of Section Modulus: Elastic and Plastic. If it is subjected to 95 KNm moment using M20 concrete and Fe 415 steel. margin-bottom: 10px; This position is known as the neutral axis. Once all the section reaches the yield strength, plastic deformation occurs across all that section. using bending equation and plotting the bending stress diagram, Skyscrapers of the world MORE BEATS FOR THE STREETS: Go bigger with 6.5 in speakers that deliver 100-Watts of crystal-clear audio. Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Overall depth of a flanged beam is equal to the sum of the depth of flange (Df) and depth of the web (dw). Do you like this page? Trying to match up a new seat for my bicycle and having difficulty finding one that will work, Poisson regression with constraint on the coefficients of two variables be the same, Comprehensive Functional-Group-Priority Table for IUPAC Nomenclature, List of resources for halachot concerning celiac disease. Framing Interior Walls| border-radius:5px; When there's plastic deformation instead of elastic deformation, we need to use the plastic section modulus. .bravenet-footer { This is essentially a weighted average of the area and distance from bottom for each segment. rev2023.1.18.43174. [Length]^4 break the section into simple geometric shapes. y_c < h-y_c e.innerHTML += 'Invalid email address'; The concrete area of the flange is to be neglected (cracked) and the beam is treated as a rectangular beam if the flange is located in the tension zone. We will take the datum or reference line from the bottom of the beam section. Calculate SF & BM for Cantilever, Deflection & Slope Calculator for Cantilever The elastic neutral axis always passes through the centroid of the cross-section and the plastic neutral axis pass through the line that divides the cross-sectional area into two parts of equal area. text-align:right; (T beam design), Critical neutral axis, n = kd = 0.29 x 500 = 145 mm. } S_x We also have some articles below about how to calculate the moment of inertia, as well as more information on centroids and section modulus. If you need more details about the procedure, get it here. This calculator is developed to help in determination of moment of inertia and other geometrical properties of plane sections of beam and column. box-sizing: border-box; The sign convention used in the shear force and bending moment diagrams is (positive values shown): Using the cursor to hover over any point on the bending moment, shear force or deflection diagrams gives the specific values at that location along the beam. Engineering Mechanical Bending stress () on beams calculator - formula & step by step calculation to find the bending stress on beams supported by the two neutral axis. solved calculate the depth of neutral axis a beam size 450x300 mm use m2s grade concrete and 4 no fe 500 steel at bottom with clear cover solved calculate the depth of neutral axis a beam size 450x300 mm use m2s grade concrete and 4 no fe 500 steel at bottom with clear cover neutral axis in beam definition formula gate notes ultimate moment { The support type can either be pinned (fixed in translation, free in rotation) or fixed (fixed in both translation and rotation) and is selected from the drop-down menu. How Do I Get Rid Of Musty Smell In Bat| Multiply the width of section 3 by `n_{3}`, Step 2:- Find position of the neutral axis. Bending Moment Calculator We can verify this result with the above free moment of inertia calculator, which shows the same result of 10.6667 in^4: Now let's look at a more complex case of where the cross section is an I beam, with different flange dimensions. The example diagram below, from the summary section, shows a two-span continuous beam with a linear distributed patch load and point load. beam experiences the highest tensile stress, Wheel speed sensor vs ABS sensor Difference, Latent heat: Definition, Formula, Types, Diagram, Example, Calculate kinetic energy without velocity? What is moment capacity. margin-bottom:10px; Please don't use computer-generated text for questions or answers on Physics, Determining the neutral axis of an I-shaped cross section with dissimilar materials. Max. These are quick moment of inertia equations that provide quick values and are a great way to cross reference or double check your results. Let your friends know about it. Calculate bending moment & shear force for simply supported beam. In the following table, we list the section modulus formula for a rectangular section and many other profiles (scroll the table sideways to see all the equations): Zx=Zy=0.25a3Z_x = Z_y = 0.25a^3Zx=Zy=0.25a3, Ix=Iy=a412I_x = I_y = \frac{a^4}{12}Ix=Iy=12a4, Sx=Sy=Ixyc=a36S_x = S_y =\frac{I_x}{y_c} = \frac{a^3}{6}Sx=Sy=ycIx=6a3, Sx=Ixyc=bd26S_x = \frac{I_x}{y_c} = \frac{b d^2}{6}Sx=ycIx=6bd2, Sy=Iyxc=db26S_y = \frac{I_y}{x_c} = \frac{d b^2}{6}Sy=xcIy=6db2, Zx=0.25(bd2bidi2)Z_x = 0.25(bd^2-b_id_i^2)Zx=0.25(bd2bidi2), Zy=0.25(db2dibi2)Z_y = 0.25(db^2-d_ib_i^2)Zy=0.25(db2dibi2), Ix=bd3bidi312I_x = \frac{bd^3-b_id_i^3}{12}Ix=12bd3bidi3, Iy=db3dibi312I_y = \frac{db^3-d_ib_i^3}{12}Iy=12db3dibi3, yc=bt2+twd(2t+d)2(tb+twd)y_c=\frac{bt^2+t_wd(2t+d)}{2(tb+t_wd)}yc=2(tb+twd)bt2+twd(2t+d), Zx=d2tw4b2t24twbt(d+t)2Z_x=\frac{d^2t_w}{4}-\frac{b^2t^2}{4t_w}-\frac{bt(d+t)}{2}Zx=4d2tw4twb2t22bt(d+t), Ix=b(d+t)3d3(btw)3A(d+tyc)2I_x = \frac{b(d+t)^3-d^3(b-t_w)}{3} \\ \ \ \ \ \ \ \ \ \ - \footnotesize A(d+t-y_c)^2Ix=3b(d+t)3d3(btw)A(d+tyc)2, Zx=t2b4twd(t+dtwd/2b)2Z_x=\frac{t^2b}{4}-\frac{t_wd(t+d-t_wd/2b)}{2}Zx=4t2b2twd(t+dtwd/2b), Iy=tb3+dtw312I_y = \frac{tb^3+dt_w^3}{12}Iy=12tb3+dtw3, Zy=b2t+tw2d4Z_y= \frac{b^2t+t_w^2d}{4}Zy=4b2t+tw2d, Sx=Ixd+tycS_x = \frac{I_x}{d+t-y_c}Sx=d+tycIx, yc=bt2+2twd(2t+d)2(tb+2twd)y_c=\frac{bt^2+2t_wd(2t+d)}{2(tb+2t_wd)}yc=2(tb+2twd)bt2+2twd(2t+d), Zx=d2tw2b2t28twbt(d+t)2Z_x=\frac{d^2t_w}{2}-\frac{b^2t^2}{8t_w}-\frac{bt(d+t)}{2}Zx=2d2tw8twb2t22bt(d+t), Ix=b(d+t)3d3(b2tw)3A(d+tyc)2I_x = \frac{b(d+t)^3-d^3(b-2t_w)}{3} \\ \ \ \ \ \ \ \ \ \ -\footnotesize A(d+t-y_c)^2Ix=3b(d+t)3d3(b2tw)A(d+tyc)2, Iy=(d+t)b3d(b2tw)312I_y = \frac{(d+t)b^3-d(b-2t_w)^3}{12}Iy=12(d+t)b3d(b2tw)3, Zx=t2b4+twd(t+dtwdb)Z_x=\frac{t^2b}{4}+t_wd(t+d-\frac{t_wd}{b})Zx=4t2b+twd(t+dbtwd), Zy=b2t4+twd(btw)Z_y= \frac{b^2t}{4} + t_wd(b-t_w)Zy=4b2t+twd(btw), Zx=twd24+bt(d+t)Z_x=\frac{t_wd^2}{4}+bt(d+t)Zx=4twd2+bt(d+t), Zy=b2t2+tw2d4Z_y= \frac{b^2t}{2}+\frac{t_w^2d}{4}Zy=2b2t+4tw2d, Ix=b(d+2t)3(btw)d312I_x = \frac{b(d+2t)^3-(b-t_w)d^3}{12}Ix=12b(d+2t)3(btw)d3, Iy=b3t6+tw3d12I_y = \frac{b^3t}{6} + \frac{t_w^3d}{12}Iy=6b3t+12tw3d, yc=d2+bt+t22(b+dt)y_c=\frac{d^2+bt+-t^2}{2(b+d-t)}yc=2(b+dt)d2+bt+t2, xc=b2+dtt22(b+dt)x_c=\frac{b^2+dt-t^2}{2(b+d-t)}xc=2(b+dt)b2+dtt2, Zx=t(dt)2b2+2bd4Z_x=t\frac{(d-t)^2-b^2+2bd}{4}Zx=t4(dt)2b2+2bd, Ix=bd3(bt)(dt)33A(dyc)2\footnotesize I_x = \frac{bd^3-(b-t)(d-t)^3}{3} \\ \ \ \ \ \ \ \ \ \ -A(d-y_c)^2Ix=3bd3(bt)(dt)3A(dyc)2, Zx=bt24+dt(dt)2t2(dt)24bZ_x= \frac{bt^2}{4}+\frac{dt(d-t)}{2}-\frac{t^2(d-t)^2}{4b}Zx=4bt2+2dt(dt)4bt2(dt)2, Iy=db3(dt)(bt)33A(bxc)2\footnotesize I_y = \frac{db^3-(d-t)(b-t)^3}{3} \\ \ \ \ \ \ \ \ \ \ -A(b-x_c)^2Iy=3db3(dt)(bt)3A(bxc)2, Sx=IxdycS_x = \frac{I_x}{d-y_c}Sx=dycIx, Sy=IybxcS_y = \frac{I_y}{b-x_c}Sy=bxcIy, Zy=t(bt)2d2+2db4Z_y=t\frac{(b-t)^2-d^2+2db}{4}Zy=t4(bt)2d2+2db, Zy=dt24+bt(bt)2t2(bt)24dZ_y= \frac{dt^2}{4}+\frac{bt(b-t)}{2}-\frac{t^2(b-t)^2}{4d}Zy=4dt2+2bt(bt)4dt2(bt)2, Zx=Zy=1.333R3Z_x = Z_y = 1.333R^3Zx=Zy=1.333R3, Ix=Iy=4R4I_x = I_y = \frac{\pi}{4}R^4Ix=Iy=4R4, Sx=Sy=Ixyc=4R3S_x = S_y =\frac{I_x}{y_c} = \frac{\pi}{4}R^3Sx=Sy=ycIx=4R3, Zx=Zy=1.333(R3Ri3)Z_x = Z_y = 1.333(R^3-R_i^3)Zx=Zy=1.333(R3Ri3), Ix=Iy=4(R4Ri4)I_x = I_y = \frac{\pi}{4}(R^4-R_i^4)Ix=Iy=4(R4Ri4), Sx=Sy=IxycS_x = S_y =\frac{I_x}{y_c}Sx=Sy=ycIx. 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