Design method of stiffening ribs in beam-column joints of steel frame structures based on overall structural stiffness

Abstract

The present invention discloses a method for designing stiffening ribs for beam-column joints of steel frame structures based on overall structural stiffness, which relates to the field of construction engineering technology and includes the following steps: performing conventional structural design on the beams and columns; calculating the main structural indicators under the condition of complete rigid connection between the beams and columns; considering the actual stiffness of the beam-column joints, obtaining the target stiffness index of the structure after stiffness reduction; obtaining a performance curve of the ring plate width-node rotational stiffness through finite element analysis of the node; performing detailed modeling of the beam-column joints, iteratively calculating the target rotational stiffness of the node; substituting the performance curve into the curve to determine the design ring plate width. By adopting this scheme, under a given structural layout, and designing according to the steps of this design method, the design width of the stiffening ring plate in the beam-column joint can be quantitatively determined. Under the premise of ensuring that the overall structural stiffness meets the engineering requirements, the node deadweight can be optimized, the material consumption can be reduced, and the node processing, construction and installation can be facilitated.

Description

Technical Field

[0001] The present invention relates to the technical field of construction engineering, and in particular to a design method for stiffening ribs of beam-column nodes of a steel frame structure based on overall structural stiffness. Background Art

[0002] Steel frame construction utilizes steel beams and columns as the primary lateral-force-resisting components to form a framework that bears the entire load of a building. Steel's high strength, light weight, and high rigidity make it suitable for constructing structures with large spans and heights. Due to its advantages, such as strong spans, light weight, excellent seismic resistance, rapid construction, and inherent compatibility with prefabricated structures, steel frame construction is widely used in high-rise buildings, including large-span structures like exhibition halls and convention centers, public transportation facilities like waiting halls, and multi-story buildings in high-intensity areas.

[0003] The overall stiffness and bearing capacity of steel frame structures depend on the connection performance at beam-column joints. Existing calculations assume a rigid beam-column connection, assuming no translational or rotational differences between beams and columns at the joints, and fully coupled degrees of freedom. However, the construction of the beam-column joint is crucial to determining the rationality and reliability of this calculation assumption. Achieving a completely rigid beam-column connection is difficult in actual engineering construction. Incompletely rigid joints can reduce joint stiffness, leading to a decrease in overall structural stiffness and compromising the structural safety.

[0004] In rigid beam-column joints, transverse stiffeners are typically installed directly opposite the beam flanges in the column body to transfer internal forces from the flanges and enhance the bending stiffness of the joint domain to ensure joint rigidity. In actual engineering projects, with the emergence of increasingly large-section columns, transverse stiffeners in the columns directly opposite the beam flanges are often constructed using a ring plate with a central opening. This not only reduces weight and controls material usage, but also facilitates the processing and welding of steel structures in the joint area. Furthermore, the holes in the ring plate can be used to route equipment pipelines such as siphon pipes. For concrete-filled steel tube columns, this is a key practice to ensure the quality of concrete pouring. The thickness of the stiffening ring plate is typically equal to or slightly thicker than the beam flange thickness by 1 to 2 mm, but there are no mature design regulations or methods for determining the quantitative value of its width. Summary of the Invention

[0005] The present invention aims to address the deficiencies of the prior art and to provide a method for designing stiffening ribs for beam-column nodes of steel frame structures based on the overall stiffness of the structure. By adopting this scheme, under a given structural layout, the design is carried out in accordance with the steps of this design method, and the design width of the stiffening ring plate in the beam-column node can be quantitatively determined. On the premise of ensuring that the overall stiffness of the structure meets the engineering requirements, the node deadweight can be optimized, the material consumption can be reduced, and the node processing and construction and installation are facilitated.

[0006] The present invention is achieved through the following technical solutions:

[0007] A method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness comprises the following steps:

[0008] Carry out structural design according to conventional design process, determine the arrangement of beams and columns and their cross-sectional parameters; the stiffeners are designed as ring plates;

[0009] In the overall structural design model, a horizontal load of an inverted triangle load pattern is applied along the floor height direction as working condition 1, and a downward vertical force is applied to the top of the column as working condition 2. The main structural indicators under the condition of complete rigid connection between the beam and column are calculated; the main structural indicators include: the first-order structural period T0, the top horizontal deformation value δ0 under working condition 1, and the linear buckling stability coefficient P under working condition 2. cr0 ;

[0010] Based on the main structural indicators and considering the actual stiffness of the beam-column joints, the target stiffness index of the structure after stiffness reduction is obtained;

[0011] A finite element analysis model of the beam-column joint was established, and the performance curve of the ring plate width-node rotational stiffness was obtained through finite element analysis of the joint.

[0012] In the overall structural design model, the beam-column joints are refined and modeled. Based on working conditions 1 and 2, the target rotational stiffness of the joints that meets the structural target stiffness index is obtained through iterative calculation.

[0013] The target node rotational stiffness is substituted into the performance curve of the ring plate width-node rotational stiffness, and the minimum ring plate width is determined based on the intersection point.

[0014] A further solution is to reduce the stiffness of the structure to target stiffness indicators including: T t =1.05T0,δ t =1.10δ0,P crt =0.9P cr0 .

[0015] A further solution is to establish a finite element analysis model of the beam-column node and obtain a performance curve of the ring plate width-node rotational stiffness through finite element analysis of the node, further comprising the following specific steps:

[0016] Model the structure according to the arrangement position of each component and its cross-sectional parameters;

[0017] After the geometric model is established, the model is meshed and assigned material properties;

[0018] Setting boundary conditions, wherein the boundary conditions include: constraining the top and bottom of the column of the node model, and the beam end is free and serves as the loading end;

[0019] Then the node rotation angle φ corresponding to the applied bending moment M is calculatedj , and according to the formula K j =M / φ j Solve the node rotation stiffness K j ;

[0020] Take the width of the ring plate as a variable and repeat the above steps to perform iterative calculations;

[0021] According to the node rotation stiffness K j , and obtain the performance curve of ring plate width-node rotation stiffness.

[0022] In a further embodiment, in the model, the total height of the steel column section is not less than twice the width of the steel column section;

[0023] The length of the steel beam section from the node end to its own free end shall not be less than the height of the steel beam section;

[0024] The ring plate is arranged opposite to the beam flange and has the same thickness as the beam flange.

[0025] A further solution is to establish the finite element analysis model of the beam-column node, and the multi-axial constitutive model of the steel adopts the classical metal plasticity model, and the uniaxial constitutive relationship adopts the double-broken line model, where Et = 0.01E0, and the steel is simulated using the S4R shell element.

[0026] A further solution is to use K j0 =35EI / L as the preset rotational stiffness, and 0.2 to 5 times K j0 Calculations and curve drawing are carried out within a certain range; where E is the elastic modulus of steel, I is the bending stiffness of the beam section, and L is the span of the beam section.

[0027] In a further embodiment, the iterative calculation based on working condition 1 and working condition 2 to obtain the target rotational stiffness of the node that meets the target structural stiffness index further includes the following specific steps:

[0028] Set the connection stiffness K j ';

[0029] Apply working conditions 1 and 2 to calculate the structural period T under different rotational stiffness j 'And the top horizontal deformation value δ corresponding to the working condition j ', linear buckling stability coefficient P under working condition 2 crj ';

[0030] Finally, take the one that satisfies T j '≤T t , δ j '≤δ t 、P crj '≥P crtThe connection stiffness K jt Serves as the target rotational stiffness of the node.

[0031] In a further solution, the step of setting the connection stiffness includes: setting a connection unit between the beam end and the column section, the translational stiffness of the connection unit is infinite, and the rotational stiffness of the connection unit is set as the connection stiffness; or setting the stiffness release at the beam end, not releasing the translational stiffness, releasing the bending stiffness, and setting the stiffness value as the connection stiffness.

[0032] A further solution is to calculate the target rotation stiffness of the node that meets the target stiffness index of the structure, and the connection stiffness K j 'Can K j0 To start with, perform a trial calculation and calculate T based on the trial calculation results. j '、δ j '、P crj 'With the target value T t , δ t 、P crt The relationship between K j ' and complete the iterative calculation; where K j0 =35EI / L.

[0033] A further solution is to round up the minimum ring plate width after determining it to obtain the final design ring plate width.

[0034] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0035] The present invention provides a design method for stiffening ribs in beam-column nodes of steel frame structures based on the overall stiffness of the structure. By adopting this scheme, under a given structural layout, the design is carried out according to the steps of the design method, and the design width of the stiffening ring plate in the beam-column node can be quantitatively determined. On the premise of ensuring that the overall stiffness of the structure meets the engineering requirements, the node deadweight can be optimized, the material consumption can be reduced, and the node processing and construction and installation are facilitated. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the following briefly introduces the drawings required for use in the examples. It should be understood that the following drawings only illustrate certain embodiments of the present invention and should not be considered as limiting the scope. A person of ordinary skill in the art can also derive other relevant drawings based on these drawings without inventive effort. In the drawings:

[0037] Figure 1 A schematic diagram of loads for working condition 1 during the overall structural analysis provided by the present invention;

[0038] Figure 2A schematic diagram of the load under working condition 2 during the overall structural analysis provided by the present invention;

[0039] Figure 3 This is an axonometric diagram of the beam-column joint analysis model provided by the present invention;

[0040] Figure 4 A schematic cross-sectional view of the beam-column node analysis model provided by the present invention;

[0041] Figure 5 A grid division diagram of the beam-column node analysis model provided by the present invention;

[0042] Figure 6 Schematic diagram of boundary conditions and moment loading of the beam-column node analysis model provided by the present invention;

[0043] Figure 7 The ring plate width-node rotation stiffness performance curve provided by the present invention;

[0044] Figure 8 This is a simplified schematic diagram of the beam-column calculation assumptions in the overall analysis model provided by the present invention;

[0045] Figure 9 Schematic diagram for determining the minimum ring plate width provided by the present invention;

[0046] Figure 10 Flowchart of the design method provided by the present invention;

[0047] Figure 11 A performance curve diagram of ring plate width-node rotation stiffness in another embodiment provided by the present invention;

[0048] Figure 12 This is a schematic diagram of determining the minimum ring plate width in another embodiment provided by the present invention.

[0049] Markings and corresponding parts names in the accompanying drawings:

[0050] 1-Steel column, 2-Steel beam, 3-Ring plate. DETAILED DESCRIPTION

[0051] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with examples and drawings. The exemplary embodiments of the present invention and their descriptions are only used to explain the present invention and are not intended to limit the present invention.

[0052] Example 1: This example 1 provides a design method for beam-column nodes of a steel frame structure based on the overall stiffness of the structure. Figures 1-10 As shown, the following specific steps are included:

[0053] 1) First, the structural design is carried out according to the conventional design process to determine the structural layout and component cross-sectional dimensions. In the overall structural design model, the horizontal load of the inverted triangle load pattern is applied along the floor height direction as the working condition 1, such as Figure 1 As shown in Figure 2, vertical force is applied to the top of the column at the top of the structure as working condition 2, as shown in Figure 2. Figure 2 As shown; then the main structural indicators under the condition of complete rigid connection of the structural beam and column are calculated, among which the main structural indicators include: the first-order structural period T0, the top horizontal deformation value δ0 under the working condition 1, and the linear buckling stability coefficient P under the working condition 2. cr0 .

[0054] After considering the actual stiffness of the nodes, the overall stiffness of the structure will be weakened. The target stiffness index of the structure after the stiffness weakening is set as follows: T t =1.05T0,δ t =1.10δ0,P crt =0.9P cr0 .

[0055] 2) Establish a finite element analysis model of typical nodes of steel structure beams and columns, such as Figure 3 As shown, a parametric analysis based on the ring plate width is performed, different node rotation stiffnesses are obtained based on different ring plate widths, and the performance curve of ring plate width-node rotation stiffness is drawn, as shown in Figure 7 shown.

[0056] The specific steps mainly include:

[0057] ① The cross-section of beams and columns is determined by the overall calculation of the project. The total height of the steel column 1 section in the model shall not be less than twice the width of the column section, and the length of the steel beam 2 section from the node end shall not be less than the height of the beam section.

[0058] ② The stiffening plate is set opposite to the beam flange, and its thickness is the same as that of the beam flange. The model is based on the ring plate 3 with a hole in the middle. The width b of the ring plate 3 is used as a variable for iterative calculation, such as Figure 4 shown.

[0059] ③ The multiaxial constitutive model of steel adopts the classical metal plasticity model (mises yield criterion, correlated flow law, isotropic strengthening); the uniaxial constitutive relationship adopts the double-broken line model, where Et = 0.01E0, and the steel plate is simulated using S4R shell elements.

[0060] ④ Such as Figure 5 As shown in the figure, after the geometric model is established, the model is meshed and material properties are assigned: the mesh size should be close to the thickness of the steel plate; the mesh should be encrypted within the range of twice the beam height outside the node area.

[0061] ⑤Finally, set the boundary conditions: constrain the top and bottom of the column in the node model, and free the beam end as the loading end, such as Figure 6 shown.

[0062] ⑥Calculate the node rotation angle φ under the corresponding bending moment j , according to formula K j =M / φ j Solve the node rotation stiffness K j ;

[0063] ⑥ After obtaining the node rotation stiffness, use the ring plate width as the horizontal coordinate and the beam rotation stiffness as the vertical coordinate, and draw the performance curve based on the existing data points, such as Figure 7 shown.

[0064] ⑦ Due to the extremely large range of node rotation stiffness, it is recommended to use K j0 =35EI / L as the preset rotational stiffness. The performance curve is drawn over a range of 0.2 to 5 times K j0 Range; where E is the elastic modulus of steel, I is the bending stiffness of the beam section, and L is the span of the beam section.

[0065] 3) In the overall structural design model, the beam-column rigid joints are modeled in detail, and the calculation assumptions are as follows: Figure 8 As shown, the translational degrees of freedom at the node of the coupled steel beam 2 and steel column 1 are set to the connection stiffness K j '; Apply the same working conditions 1 and 2 in step 1) to calculate the structural period T under different rotational stiffness j ' and the top horizontal deformation value δ' corresponding to working condition 1, the linear buckling stability coefficient P under working condition 2 crj ', take the one that satisfies T j '≤T t , δ j '≤δ t 、P crj '≥P crt The connection stiffness K jt Serves as the target rotational stiffness of the node.

[0066] The above steps also include: ① There are usually two methods in the software to achieve the setting of the connection stiffness of the rotational degree of freedom: one is to set a connection unit between the beam end and the column segment, set the translation stiffness of the connection unit to infinity, and set the rotation stiffness of the connection unit to the connection stiffness; the other is to set the stiffness release at the beam end, not release the translation stiffness, release the bending stiffness, and set the stiffness value to the connection stiffness;

[0067] ② Due to the extremely large range of node rotation stiffness, calculate the connection stiffness K j 'It is recommended to use K j0 To start with, perform a trial calculation and calculate T based on the trial calculation results. j '、δ j '、Pcrj 'With the target value T t , δ t 、P crt The relationship between K j 'adjust the value and complete the iterative trial calculation to reduce the workload, where K j0 =35EI / L.

[0068] 4) If Figure 9 As shown, the node target rotation stiffness K is drawn in the performance curve in step 2). jt The horizontal line of the performance curve corresponds to the horizontal abscissa b min This is the minimum ring plate width, which is rounded up to get the designed ring plate width.

[0069] In the above design method, the stiffness requirement of the actual calculated structure is used to quantitatively propose the node stiffness requirement for the specific structure, which is targeted and quantifiable. The design method uses the first-order natural vibration period T of the structure, the vertex displacement δ under the inverted triangle load mode, and the buckling stability factor P under the top vertical load. Cr As indicators for measuring the overall stiffness of the structure, the changes in these three indicators at different node stiffnesses are used to measure the overall structural stiffness degradation caused by the nodes not being ideally rigidly connected but having rotational stiffness, and recommended control limits are proposed.

[0070] The above design method establishes the connection between the node stiffener and the node rotation stiffness, and the node stiffness under the specified width structure can be quantitatively obtained through finite element analysis.

[0071] By combining the above measures, an executable and easy-to-implement design method for the width of the center column stiffener in the beam-column joint of the steel frame structure based on the overall stiffness of the structure was established.

[0072] In addition, the connection nodes between steel tube concrete columns and steel beams can also be analyzed. The main differences and key points are: steel tube concrete columns are composed of an outer steel tube and embedded concrete. In the model establishment step, it is noted that the plastic damage model is used for the concrete material constitutive model, the concrete element is simulated using C3D8R solid elements, the ring plate stiffeners are embedded in the concrete using embedded region constraints, and the concrete and steel column side walls adopt surface-to-surface contact: the hard contact model is used for normal contact, and the Coulomb friction mechanics model is used for tangential contact.

[0073] This patented method can also be used to analyze the steel column and steel beam connection nodes of the steel frame + core tube structure. The difference is that the preset target rotational stiffness value in steps 2) and 3) is adjusted to 15EI / L during the trial calculation process, which can improve the efficiency of the trial calculation analysis.

[0074] Example 2: This example is an application example of Example 2 based on Example 1. Figure 11 and Figure 12 As shown, it is applied in a steel structure frame project.

[0075] 1. Complete the conventional design process in 3d3s software and determine the typical layout. The beam cross-section is welded H-beam H1000X300X12X30, with a typical span of 13 meters.

[0076] 2. The calculated first-order structural period T0 = 0.91S, the top lateral displacement δ0 = 25mm under working condition 1, and the linear buckling stability coefficient P under working condition 2 cr0 =95.

[0077] 3. Set the target structural stiffness index to: T t =1.05x0.91=0.96s;δ t =1.10x25=27.5mm;P crt =0.9x95=85.5.

[0078] 4. Establish a finite element calculation model for the beam-column node. Starting from the ring plate width b = 100 mm, increase the ring plate width in steps of 50 mm until the ring plate width b = 350 mm, and calculate the corresponding node rotation stiffness K. j , plot the ring plate width b-node rotation stiffness K j Performance curves such as Figure 11 shown.

[0079] 5. Set the end unit rotational stiffness release in the 3d3s software, and set the stiffness of the steel beam's i / j node (corresponding to the two ends) around the 3rd axis (strong axis, corresponding to the vertical bending moment direction). Initially, the estimated stiffness index is 35EI / L = 2.81e9 kN·m / rad. The calculated stiffness index does not meet the target structural stiffness index, so the node stiffness is increased for further trial calculations. When the calculated stiffness result meets the target structural stiffness index, the required node rotational stiffness is obtained to be 3.42e9 kN·m / rad.

[0080] 6. Such as Figure 12 As shown, the b-node rotation stiffness K at the ring plate width is j Draw a horizontal line on the performance curve for the required node rotation stiffness of 3.42e9 kN·m / rad and determine the minimum ring plate width b corresponding to the intersection position. min =165mm, the design is rounded up to finally determine the ring plate width b to be 200mm.

[0081] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for designing stiffening ribs for beam-column joints of steel frame structures based on overall structural stiffness, characterized in that: The following steps are involved: Carry out structural design according to conventional design process, determine the arrangement of beam and column components and their cross-sectional parameters; the stiffening rib is designed as a ring plate (3); In the overall structural design model, a horizontal load of an inverted triangle load pattern is applied along the floor height direction as working condition 1, and a downward vertical force is applied to the top of the column as working condition 2. The main structural indicators under the condition of complete rigid connection between the beam and column are calculated; the main structural indicators include: the first-order structural period T0, the top horizontal deformation value δ0 under working condition 1, and the linear buckling stability coefficient P under working condition 2. cr0 ; Based on the main structural indicators and considering the actual stiffness of the beam-column joints, the target stiffness index of the structure after stiffness reduction is obtained; A finite element analysis model of the beam-column joint was established, and the performance curve of the ring plate width-node rotational stiffness was obtained through finite element analysis of the joint. In the overall structural design model, the beam-column joints are refined and modeled. Based on working conditions 1 and 2, the target rotational stiffness of the joints that meets the structural target stiffness index is obtained through iterative calculation. The target node rotational stiffness is substituted into the performance curve of the ring plate width-node rotational stiffness, and the minimum ring plate width is determined based on the intersection point.

2. A method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 1, characterized in that: The target stiffness indexes of the structure after stiffness reduction include: T t =1.05T0,δ t =1.10δ0,P crt =0.9P cr0 .

3. The method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 1, characterized in that: The establishment of the beam-column node finite element analysis model and obtaining the performance curve of the ring plate width-node rotational stiffness through the node finite element analysis also includes the following specific steps: Model the structure according to the arrangement position of each component and its cross-sectional parameters; After the geometric model is established, the model is meshed and assigned material properties; Setting boundary conditions, wherein the boundary conditions include: constraining the top and bottom of the column of the node model, and the beam end is free and serves as the loading end; Then the node rotation angle φ corresponding to the applied bending moment M is calculated j , and according to the formula K j =M / φ j Solve the node rotation stiffness K j ; The width of the ring plate (3) is used as a variable and the above steps are repeated for iterative calculation; According to the node rotation stiffness K j , and obtain the performance curve of ring plate width-node rotation stiffness.

4. A method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 3, characterized in that: In the model, the total height of the steel column (1) section is not less than twice the width of the steel column (1) section; The cross-sectional length of the steel beam (2) from the node end to its own free end shall not be less than the cross-sectional height of the steel beam (2); The ring plate (3) is arranged opposite to the beam flange and has the same thickness as the beam flange.

5. The method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 3, characterized in that: When establishing the finite element analysis model of the beam-column node, the multiaxial constitutive model of steel adopts the classical metal plasticity model, the uniaxial constitutive relationship adopts the double-broken line model, where Et=0.01E0, and the steel plate is simulated using the S4R shell element.

6. The method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 3, characterized in that: When drawing the performance curve of ring plate width-node rotation stiffness, K j0 =35EI / L as the preset rotational stiffness, and 0.2 to 5 times K j0 Calculations and curve drawing are carried out within a certain range; where E is the elastic modulus of steel, I is the bending stiffness of the beam section, and L is the span of the beam section.

7. The method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 1, characterized in that: The iterative calculation based on the first and second working conditions to obtain the node target rotational stiffness that meets the structural target stiffness index also includes the following specific steps: Set the connection stiffness K j '; Apply working conditions 1 and 2 to calculate the structural period T under different rotational stiffness j 'And the top horizontal deformation value δ corresponding to the working condition j ', linear buckling stability coefficient P under working condition 2 crj '; Finally, take the one that satisfies T j '≤T t , δ j ' ≤δ t 、P crj '≥P crt The connection stiffness K jt Serves as the target rotational stiffness of the node.

8. The method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 7, characterized in that: The step of setting the connection stiffness includes: setting a connection unit between the beam end and the column section, the translation stiffness of the connection unit is infinite, and the rotation stiffness of the connection unit is set as the connection stiffness; or setting stiffness release at the beam end, not releasing the translation stiffness, releasing the bending stiffness, and setting the stiffness value as the connection stiffness.

9. The method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 7, characterized in that: When calculating the target rotation stiffness of the node that meets the target stiffness index of the structure, the connection stiffness K j 'K j0 To start with, perform a trial calculation and calculate T based on the trial calculation results. j ', δ j ' 、P crj 'With the target value T t , δ t 、P crt The relationship between K j ' and complete the iterative calculation; where K j0 =35EI / L.

10. The method for designing stiffening ribs for beam-column joints of a steel frame structure based on overall structural stiffness according to claim 1, characterized in that: After determining the minimum ring plate width, round it up to get the final designed ring plate width.

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